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PLANE AND SPHERICAL 
TRIGONOMETRY 

(WITH FIVE-PLACE TABLES) 



A TEXT-BOOK FOR TECHNICAL SCHOOLS 
AND COLLEGES 



BY 
ROBERT E. MORITZ, Ph.D., Ph.N.D, 

PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF WASHINGTON 



NEW YORK 

JOHN WILEY & SONS, Inc. 

London: CHAPMAN & HALL, Limited 

1913 



Q/\53l 

.Hi 



Copyright, 1913, 

BY 

ROBERT E. MORITZ 



Stanbopc fliress 

F. H.GILSON COMPANY 
BOSTON, U.S.A. 



/^ ~, . ~ — ~ 



PREFACE 



In preparing the present volume the author has tried to give proper 
recognition to the fundamental and far-reaching changes that have 
been wrought during the last decade in the teaching of the elementary 
branches of mathematics and to bring about a more perfect adjust- 
ment of the teaching of trigonometry with the teaching of algebra and 
geometry on which it rests and with the progress of the arts and 
sciences to which it applies. Among the distinctive features of the 
present volume the following deserve special emphasis. 

i. The subject is introduced with a chapter on the graphic solution 
of triangles which shows the immediate connection of trigonometry 
with geometry. Similarly the six cases of spherical triangles are 
solved by the graphic method before the analytical methods are 
presented. 

2. The references to algebra are limited to those with which every 
beginner may be reasonably assumed to be familiar. Other principles 
are developed as they are needed. 

3. A knowledge of logarithms has not been presupposed. For 
this reason a chapter on logarithms and their use has been incorporated 
in the text. 

4. Many of the fundamental formulas and theorems have been 
derived by two or more independent methods. 

5. Functions of acute, obtuse and general angles are treated in 
separate chapters. 

6. The study of the general angle and the analytical derivation of 
geometrical formulas and theorems has been placed after the solution 
of triangles. This makes it possible for the student to complete the 
subject through the solution of plane triangles in a six weeks' course 
of four or five recitations per week. 

7. The student is trained in accuracy and self-confidence in numeri- 
cal calculations by a constant insistance on proper checks and veri- 
fication of results. 



VI PREFACE 

8. The accuracy of results is strictly limited to that of the data 
employed. The student is constantly guarded against superfluous 
figures and a show of accuracy not warranted by the data or the 
processes of computation employed. 

9. The student is guarded against disregarding figures and re- 
mainders without first determining the effect of the parts neglected 
on the final results. Thus, in the computation of logarithms by means 
of the logarithmic series, or of natural functions by means of the sine 
and cosine series, the effect of the neglected part of each series on the 
final result has been carefully considered in each case. 

10. The student is led by easy stages from simple to complicated 
computations. For this purpose each of the cases of both plane and 
spherical triangles is followed by sets of problems involving suc- 
cessively three, four and five significant figures. This enables the 
student to become familiar with his formulas before he enters on the 
more exacting computations. 

11. The best practice of the government computing offices, and of 
practical engineers has been followed in the arrangement of the 
logarithmic work. 

12. Trigonometric curves have received much fuller treatment 
than is usual. The treatment includes curves of any given amplitude 
and wave-length, logarithmic and exponential curves, composition of 
harmonic curves, the catenary and the curve of damped vibrations. 

13. A special section is devoted to the angle and its functions 
considered as functions of time. 

14. The trigonometric representation of imaginary and complex 
numbers is developed in connection with the theorems of Euler and 
Demoivre and the development of the functions in series. 

15. Hyperbolic functions have received much fuller treatment than 
is usual. The analogies between the circular and hyperbolic functions 
have been developed both analytically and geometrically. 

16. It is believed that a rigorous proof of Napier's Rules of Circular 
Parts appears here for the first time in an elementary text. 

17. It is believed that the simultaneous derivation of the three 
fundamental formulas for the spherical triangle (11, 36) has not 
previously appeared in a textbook on trigonometry. 

18. The lists of applied problems will be found more complete 
than any that have appeared in recent American texts. The prob- 
lems are segregated into sets relating separately to Physics, Engi- 



PREFACE 



VU 



neering, Navigation, Geography, Astronomy and Plane and Solid 

Geometry. 

19. Abundant historical matter has been introduced throughout 

the text. 

ROBERT E. MORITZ. 

September, 19 13. 



GREEK ALPHABET 



a pronounced alpha. 



(8 


' beta. 


7 


' gamma. 


5 


delta. 


6 


epsilon. 


r 


' zeta. 


V 


1 eta. 


e ' 


1 theta. 


1 


' iota. 


K 


kappa. 


X 


1 lambda. 


M 


' mu. 



V 


pronounced nu. 

XI. 





a 


omicron 


7T 
P 


it 


pi. 
rho. 


r 


a 

a 


sigma. 
tau. 


V 

<$> 
X 


it 
tt 
u 


upsilon. 

phi. 

chi. 


4> 


tt 


psi. 


0} 


it 


omega. 



CONTENTS 



PART I. PLANE TRIGONOMETRY 

CHAPTER I 

Introduction 

Art. Page 

i. Graphic solution of triangles i 

2. Solution of practical problems by the graphic method 4 

3. Inadequacy of the graphic method 7 

4. Definition of trigonometry 7 

CHAPTER II 

Trigonometric Functions of an Acute Angle 

5. Definition of function 9 

6. Definition of reciprocal 9 

7. The six trigonometric functions of an acute angle 9 

8. Trigonometric functions determined approximately by measurement .... 14 

9. Given one of its functions, to construct the angle 17 

10. Functions of complementary angles 19 

11. Functions of o°, 30 , 45 , 6o°, 90 21 

12. Fundamental relations 24 

13. To express each of the functions in terms of a given one 25 

14. Reduction of trigonometric expressions to their simplest form 27 

15. Trigonometric identities 33 

CHAPTER III 

Solution oe Right Triangles by Natural Functions 

16. Table of natural functions 35 

17. To find the natural functions of an angle less than 90 35 

18. To find the angle less than 90 corresponding to a given natural function 39 

19. Accuracy of results 43 

20. Solution of right triangles by natural functions 45 

21. Triangles having a small angle 49 

22. Historical note 50 

23. Review 51 

CHAPTER TV 

Logarithms 

24. Definition of logarithm $$ 

25. Fundamental laws governing logarithms 54 

26. Logarithms of special values , 55 

ix 



X CONTENTS 

Art. Page 

27. The common system of logarithms 57 

28. Rule for the characteristic 58 

29. Table of common logarithms 61 

30. To find the logarithm of a given number 62 

31. To find the number corresponding to a given logarithm 65 

32. Directions for the use of logarithms 67 

33. Application of logarithms 72 

34. To compute a table of common logarithms 73 

35. Relation between log a N and log^N 75 

36. Natural or hyperbolic logarithms 76 

37. Tables of logarithmic trigonometric functions 78 

38. To find the logarithmic trigonometric functions of an angle less than 90 79 

39. To find the angle corresponding to a given logarithmic trigonometric 

function 81 

40. Logarithmic functions of angles near o° or 90 83 

41. Use of S and T table 84 

42. Historical note 87 



CHAPTER V 

Logarithmic Solution of Right Triangles 

43. Logarithmic solution of right triangles 89 

44. Number of significant figures 93 

45. Applied problems involving right triangles 94 

46. Heights and distances ' 95 

47. Problems for engineers 96 

48. Applications from physics 98 

49. Problems in navigation 100 

50. Geographical and astronomical problems 103 

51. Geometrical applications 106 

52. Oblique triangles solved by right triangles no 



CHAPTER VI 

Functions of an Obtuse Angle 

53. Rectangular coordinates 117 

54. Definitions of the trigonometric functions of any angle less than 180 .... 118 

55. The signs of the functions of an obtuse angle 119 

56. Fundamental relations 119 

57. Functions of supplementary angles 120 

58. Functions of (90 -\- 6) 121 

59. Functions of 180 121 

60. Angles corresponding to a given function 122 

61. Review 123 



CONTENTS XI 

CHAPTER VII 

Properties oe Triangles 

Art. Page 

62. The law of sines 125 

63. The projection theorem 126 

64. The law of cosines 127 

65. Arithmetic solution of triangles 128 

66. The law of tangents 130 

67. Formulas for the area of a triangle 132 

68. Functions of half the angles in terms of the sides 134 



CHAPTER VIII 

Solution of Oblique Triangles 

69. Solution of oblique triangles 138 

70. Case I. Given two angles and one side 138 

71. Case II. Given two sides and the angle opposite one of them 141 

72. Case III. Given two sides and the included angle 144 

73. Case IV. Given three sides 147 

74. Practical applications 150 

(a) System of triangles 150 

(b) Auxiliary geometrical constructions 152 

(c) System of simultaneous equations 155 

75. Miscellaneous heights and distances 158 

76. Applications from physics 161 

77. Applications from surveying and engineering 164 

78. Applications from navigation 171 

79. Problems from astronomy and meteorology 172 

80. Geometrical applications 174 

CHAPTER IX 

The General Angle and its Measures 

81. General definition of an angle 177 

82. Positive and negative angles 178 

83. Complement and supplement 178 

84. Angles in the four quadrants 178 

85. Sexagesimal measure of angles 179 

86. Decimal division of degrees 180 

87. Centesimal measure of angles 180 

88. The circular or natural system of angular measures 181 

89. Comparison of sexagesimal and circular measure 182 

90. Relation between angle, arc and radius 185 

90a. Area of circular sector 187 

91. Review 189 



jdi CONTENTS 

CHAPTER X 

Functions or any Angle 

Art. Page 

92. Definition of the trigonometric functions of any angle 191 

93. Signs of the functions in each of the quadrants 192 

94. Periodicity of the trigonometric functions 193 

95. Changes in the value of the functions 193 

96. Changes in the value of the tangent 195 

97. Summary of results 195 

98. Fundamental relations 196 

99. Representation of trigonometric functions by lines. 196 

100. Reduction of the functions to the first quadrant 199 

iot. Reductions from the third quadrant 199 

102. Reductions from the fourth quadrant 201 

103. Functions of negative angles 203 

104. Table of principal reduction formulas and general rules 204 

105. Generalization of the preceding reduction formulas 206 



CHAPTER XI 

Functions of Two or More Angles 

106. Addition theorem for the sine and cosine 209 

107. Generalization of the addition theorems 210 

108. Addition theorems. Second proof 211 

109. Subtraction theorems for the sine and cosine • 212 

no. Tangent of the sum and difference of two angles 215 

in. Functions of double an angle 216 

112. Functions of half an angle 216 

113. Sums and differences of sines or cosines transformed into products. ... 219 



CHAPTER XII 

Trigonometric Equations 

114. Angles corresponding to a given function 225 

115. Principal value 225 

116. Formula for angles having a given sine 226 

117. Formula for angles having a given cosine 226 

118. Formulas for angles having a given tangent 227 

119. Summary of results 227 

120. Trigonometric equations involving a single angle 227 

121. Trigonometric equations involving multiple angles 233 

122. Trigonometric equations involving two or more variables 236 

123. Solutions adapted to logarithmic computation 240 

1 24. Inverse functions 245 

125. Review 250 



CONTENTS Xlil 
CHAPTER XIII 

Trigonometric Curves 

Art. Page 

126. Functions represented by curves 253 

127. The straight line 253 

128. The circle 254 

129. The hyperbola 255 

130. The sine curve 255 

131. The tangent curve 257 

132. The sinusoidal or simple harmonic curves 260 

133. Angles as functions of time 262 

134. Composition of sinusoidal curves 264 

135. Theorem on composition of sinusoidal curves having equal wave 

lengths 267 

136. Fourier's theorem 268 

137. The logarithmic curve 269 

138. The exponential curve 270 

139. The general exponential curve 270 

140. The compound interest law 272 

141. The catenary 274 

142. The curve of damped vibrations 275 



CHAPTER XIV 

Trigonometric Representation of Complex Numbers 

143. Imaginary numbers 278 

144. Geometric representation of imaginary numbers 278 

145. Geometric representation of complex numbers 280 

146. Trigonometric representation of complex numbers 281 

147. Geometric addition and subtraction of complex numbers 282 

148. Physical applications of complex numbers 283 

149. Historical note 285 

150. Multiplication and division of complex numbers 287 

151. Powers of complex numbers 288 

152. Roots of complex numbers 289 

153. To solve the equation z n — 1 = 292 

.154. To solve the equation z n + 1 = o 293 

155. The cube roots of unity 295 

156. The cube roots of any real or complex number 295 

157. Solution of cubic equations 296 

158. The irreducible case 298 

159. To express sin nd and cos nd in powers of sin and cos . 301 

160. To express cos and sin in terms of sines and cosines of multiple 

angles 302 



XIV CONTENTS 

CHAPTER XV 

Trigonometric Series and Calculation of Tables 

Art. Page 

161. Definition of infinite series 306 

162. Convergent and non- convergent series 307 

163. Absolutely convergent series 308 

164. Sum of an infinite series 310 

165. The limit or r n as n approaches infinity 310 

166. The geometric infinite series 310 

167. Convergency test 311 

168. Convergency of special series 313 

169. The number e 316 

170. The exponential series 318 

171. The logarithmic series 320 

172. Calculations of logarithms 321 

173. Errors resulting from the use of logarithms 324 

174. Limiting values of the ratios > ■ , as x approaches zero 326 

x x 

x /sin (x/n)\ n 

175. Limiting values of cos n - and ( j- — ) as n approaches infinity.. . . 328 

n \ x/n ) 

176. The sine, cosine and tangent series 329 

177. Computation of natural functions table 330 

178. Approximate equality of sine, tangent and radian measure of small 

angles 333 



CHAPTER XVI 

Hyperbolic Functions 

179. Series with complex terms 336 

180. Definition of the trigonometric functions of complex numbers. ....... 337 

181. Euler's theorem, e id = cos 6 + i sin 338 

182. Geometrical representation of Euler's theorem 339 

183. Exponential form of the sine and cosine 340 

184. Hyperbolic functions defined 342 

185. Duality of circular and hyperbolic functions 343 

186. Table of formulas 345 

187. Inverse hyperbolic functions 34& 

188. Geometrical representation of hyperbolic functions 349 

189. Area of hyperbolic sector 350 

190. Use of hyperbolic functions 353 

191. Review 354 



CONTENTS XV 

PART II. SPHERICAL TRIGONOMETRY 

CHAPTER I 

Introduction 
Art. Page 

i. Definition of spherical trigonometry i 

2. The uses of spherical trigonometry i 

3. Spherical trigonometry dependent on solid geometry 2 

4. Classification of spherical triangles 3 

5. Co-lunar triangles 4 

6. Use of co-lunar triangles 4 

7. Polar triangles 6 

8. The six cases of spherical triangles . . . 7 

9. Solution of spherical triangles 7 

10. The use of the polar triangle 8 

n. Construction of spherical triangles 9 

12. The general spherical triangle 11 

CHAPTER n 
Right and Quadrantal Spherical Triangles 

13. Formulas for right spherical triangles 14 

14. Plane and spherical right triangle formulas compared 16 

15. Generalization of the right triangle formulas 16 

16. Napier's rules of circular parts 17 

17. Proof of Napier's rules of circular parts 18 

18. To determine the quadrant of the unknown parts .- 20 

19. The ambiguous case of right spherical triangles 21 

20. Solution of right spherical triangles 21 

21. Solution of quadrantal triangles 25 

22. Formulas for angles near o°, 90 , 180 26 

23. Oblique spherical triangles solved by the method of right triangles 28 

CHAPTER in 

Properties of Oblique Spherical Triangles 

24. The law of sines ^^ 

25. The law of cosines 34 

26. Relation between two angles and three sides 35 

27. Analytical proof of the fundamental formulas 36 

28. Fundamental relations for polar triangles 37 

29. Arithmetic solution of spherical triangles 37 

30. Formulas of half the angles in terms of the sides 39 

31. Formulas of half the sides in terms of the angles 41 

32. Delambre's (or Gauss's) proportions 43 



xvi CONTENTS 

Art. Page 

33. Napier's proportions 44 

34. Formulas for the area of a spherical triangle 45 

35. Plane and spherical oblique triangle formulas compared 47 

36. Derivation of plane triangle formulas from those of spherical triangles. ... 48 

CHAPTER IV 
Solution of Oblique Spherical Triangles 

37. Preliminary observations 51 

38. Case I. Given the three sides 51 

39. Case II. Given the three angles 53 

40. Case III. Given two sides and the included angle 55 

41. Case IV. Given two angles and the included side 56 

42. Case V. Given two sides and the angle opposite one of them 58 

43. Case VI. Given two angles and the side opposite one of them 60 

44. To find the area of a spherical triangle 62 

45. Applications to geometry 62 

46. Applications to geography and navigation 64 

47. Applications to astronomy 66 



PLANE TRIGONOMETRY 



CHAPTER I 

INTRODUCTION 

In order to work the exercises in this chapter the student should be provided 
with a pair of compasses, a protractor, and a graduated ruler divided into tenths 
of a unit. 

1. Graphic Solution of Triangles. In plane geometry it is shown 
that the six parts (three sides and three angles) of any plane triangle 
are so related that any three parts suffice to determine the shape of 
the triangle, and if one of the known parts is a side, the size of the 
triangle is also determined. Furthermore it is shown how to con- 
struct the triangle when a sufficient number of parts is given. All 
possible cases come under one or another of the following four cases. 

To construct the triangle when there is given, — 

I. One side and two angles. 

II. Two sides and an angle opposite one of them. 

III. Two sides and the included angle. 

IV. Three sides. 

Usually we have given not the actual lines and angles but their 
measures. From these measures lines .and angles corresponding to 
the actual lines and angles may then be constructed by means of 
suitable instruments. Such instruments are,— 

i. A graduated straight-edge for the construction and measure- 
ment of straight lines of definite lengths. The smallest divisions of 
the straight-edge should be decimal, either millimeters or tenths of 
an inch. 

2. A pair of compasses for the construction of circles and cir- 
cular arcs. 

3. A protractor for the construction and measurement of plane 
angles of definite magnitudes. 



PLANE TRIGONOMETRY 



[chap, r 




Example i. It is required to construct a triangle which has two 
sides equal to 2.5 inches and 1.75 inches respectively and the included 
angle equal to 3 6°. 

Solution. By means of the protractor 
construct an angle MAN (Fig. 1) equal 
to 36 . On AM measure off AB equal 
to 2.5 inches. On AN measure off AC 
equal to 1.75 inches. Join B and C by 
a straight line. ABC is the required 
triangle. 

The numerical values of the parts which were not known at the 
outset may now be found by measurement. BC is thus found to 
be 1.49 inches, and by means of the protractor, angles B and C are 
found to be approximately 43. 5 and 100. 5 respectively. 

If it is not possible or convenient to construct the triangle full size, 
a similar triangle may be constructed on a reduced scale; that is, any 
unit or a fraction of a unit on the scale may be taken to represent 
any unit occurring in the problem. Thus lines 3 and 4 inches long 
may be employed in the solution of a triangle whose sides are 30 and 
40 miles respectively. The angles of the reduced triangle will of 
course be equal to the angles of the triangle represented. 

Similarly, the unknown parts of a triangle which is too small for 
actual construction, say some microscopic triangle, may be found 
by measurement from a similar triangle drawn on an enlarged 
scale. 

Example 2. One side of a triangle measures 600 miles, and the 
angles adjacent to this side measure 23 ° and ioo° respectively. 
Find the remaining parts of the triangle. 

Solution. Let § inch represent 100 
miles. Then a line AB drawn 3 inches 
long will represent 600 miles. At A and 
B draw the angles BAC and ABD 23 
and ioo° respectively. Let BD intersect 
AC at E. ABE will represent the re- 
quired triangle. 

Angle E measures 57 , which of course could have been found 
otherwise by subtracting the sum of the angles A and B from 180 . 
AE and BE are found to measure approximately 3.52 and 1.40 
inches respectively. Remembering that each J inch represents 100 




i] INTRODUCTION 3 

miles, the actual lengths represented by AE and BE are approxi- 
mately 704 and 280 miles respectively. 

Solutions, like the foregoing, in which geometrical drawings to a 
scale are employed instead of numerical calculations, are called 
graphic solutions. 

Exercise i 

1. Review the following propositions in geometry. A, B, C repre- 
sent the three angles of any triangle and a, b, c the sides opposite 
these angles. 

a. Given A, B, c; to construct the triangle. 

b. Given a, b, C; to construct the triangle. 

c. Given a, b, c; to construct the triangle. 

d. Given a, b, A ; to construct the triangle. 

e. Under what conditions will (d) give rise to two different 

solutions? To only one solution? 

The following problems are to be solved by the graphic method.* 

2. Given a = 5, b = 4, c = 7; find the angles to the nearest 15'. 

Ans. A = 44 30', B = 34 , C = 101 30'. 

3. Given b = 4, c = 5, C— 90 ; find the third side and the angles 
to the nearest 15'. Ans. a = 3, A = 37 , B = 53 . 

4. Given b— 270, c = 600, A = ioo°; find the third side correct 
to the nearest integer. Ans. a = 700. 

5. Given a = 0.029, B = 32 15', C= 136 45'; find the remaining 
sides. Ans. b = 0.081, c = 0.104. 

6. Given a = 42, b = 51, A = 55 ; find the approximate measures 
of the remaining parts. Ans. c = 33.6, B = 84 , C= 41 ; 

or c= 24.9, B = 96 , C = 29 . 

7. Given A = 44 30', B = 57 , C = 78 30'; find the ratios between 
the sides opposite these angles. 

Ans. Approximately a : b : c = 5 : 6 : 7. 

* In order to employ the graphic method successfully the student must prac- 
tice accuracy. Two pencils of medium hardness should be used, one sharpened 
to a point for marking distances, the other sharpened like a chisel for drawing 
lines. The pencil points are easily kept sharp with the aid of a piece of fine sand- 
paper. The lines should be drawn sharply and they should bisect the points 
through which they are intended to pass. In measuring the required parts, begin- 
ners should estimate angles to quarters of a degree and lengths to quarters of 
the smallest division of the scale. 



PLANE TRIGONOMETRY 



[chap. I 




Fig- 3- 



2. Solution of Practical Problems by the Graphic Method. 

Many important practical problems, in which a high degree of 
accuracy is not essential, can be easily solved by the graphic method. 
Suppose it is required to find the approximate distance AB across a 
lake or'swamp, without actually measuring it. This may be accom- 
plished in various ways, one of which is as follows: 

Select some point P from which both 
A and B are visible, and measure the dis- 
tances AP and BP and also the angle 
APB* This gives two sides and the in- 
cluded angle of the triangle APB from 
which AB may be found by the method of 
the preceding article. 

Similarly the heights of towers and trees 
and mountains, of clouds and shooting 
stars, the distances through impenetrable forests across swamps and 
through mountains, the widths of rivers, ravines and canyons, may be 
determined. Even the distances between celestial objects may be 
approximated by the graphic method after certain other distances 
and angles have been measured. 

Example i. In order to determine the width of a river, the dis-, 
tance between two points A and B close to the bank of the river was' 
measured and was found to be 600 feet. The angles BAP and 
ABP, formed with a point P close to the opposite bank of the river, 
were also measured and were found to be 50 and 3 6° respectively. 
Required the approximate width of river. 

Solution. Select a suitable scale, say 1 inch 
to 100 feet, and construct a triangle ABP, 
having AB = 6 inches and the adjacent 
angles equal to 50 and 36 respectively. From 
P draw PT perpendicular to AB. PT will 
represent the width of the river. Measure 
PT. PT will be found to measure 2.7 inches, and since each inch 
represents 100 feet, the width of the river is 270 feet. 

Definitions. Let P be any point and the position of the 

observer. Through P draw a vertical line, and through draw 

a horizontal line meeting the vertical line in H. 

* The angle between two visible objects is readily measured by means of an 
instrument called a transit. 




Fig. 4. 



INTRODUCTION 



5 




Fig. 5- 



If P is above H, as in the upper figure, the angle HOP is called the 
angle of elevation of the point P as seen from O. 

If P is below H, as in the lower figure, the 
angle HOP is called the angle of depression of the . 
point P as seen from O. , 

It is obvious that the angle of elevation or 
of depression of an object depends upon the 
position of the observer. 

Example 2. From a point P at the foot of a mountain, the angle 
of elevation of the summit M is measured and is found to be 30 ; 
after walking two miles toward the summit on an incline averaging 
15°, the angle of elevation is found to measure 45 . Required the 

height of the mountain. 

Solution. Draw a horizon- 
tal fine PX. Construct an 
angle XPN = 30 ; then PN 
represents the direction in 
which the summit of the 
mountain is seen from P. 
Construct angle XPC = 15°, 
and take PC two units in 
length. Then, if each unit 
represents one mile, C will represent the position from which the 
second observation was made. 

Through C draw CX r parallel to PX, and construct an angle X f CN' 
— 45 . Then CN r represents the direction in which the summit is 
seen from C. 

Since the summit is on each of the lines PN and CN', it must be 
located at their point of intersection M. 

Draw MF perpendicular to PX. Then MF will represent the 
height of the mountain on the same scale on which PC represents two 
miles. Measure MF. If PC was taken equal to 6 inches, MF will 
measure 5.8 inches. Since 3 inches represents one mile, MF rep- 
resents 1.933 miles, or 10,200 feet approximately. 

Exercise 2 

The following problems are to be solved graphically. The stu- 
dent is expected to- obtain distances correct to three figures and 
angles correct to nearest 15'. 




Fig. 6. 



6 PLANE TRIGONOMETRY [chap, i 

i. At a distance of 400 feet from the foot of a tree, the top of 
the tree subtends an angle of 20 . Find the height of the tree. 

Ans. 145.6 ft. 

2. A straight road leads from a town A to a town B 8 miles dis- 
tant; another road leads from A to a third town C 10 miles distant. 
The angle between the roads is 65 . How far is it from B to C ? 

Ans. 9.82 mi. 

3. What is the altitude (= angle of elevation) of the sun, when a 
building 75 feet high casts a shadow 190 feet long on a horizontal 
plane? Ans. 21 30'. 

4. The great pyramid of Gizeh is 762 feet square at its base 
and each face makes an angle of 5i°5i' with the horizontal plane. 
Determine the height of the pyramid, assuming that it comes to 
an apex. Ans. 485 feet. 

5. As a matter of fact, the pyramid mentioned in Problem 4 does 
not come to a point, but terminates in a platform 32 feet square. 
Find the actual height of the pyramid. Ans. 465 feet. 

F 6. An observer on board ship sees two headlands in a straight 
line N. 35 E. The ship sails northwest for 5 miles, when one of the 
headlands appears due east and the other due northeast. How far 
apart are the headlands? 

7. Two observers on opposite sides of a balloon observe the balloon 

at the same instant and find its angles of elevation to be 56 and 42 

respectively. The observers are one mile apart. Find the height of 

the balloon at the time the observations were taken. 

Ans. 0.6 mi. nearly. 

8. In order to determine the distance across a swamp, a distance 
AB was laid off 100 yards long, and at each extremity of the line AB 
the angles were measured between the other extremity of the line 
and each of two stakes P and Q placed at opposite ends of the swamp. 
At one extremity of the line the angles measured 35 and 85 respec- 
tively, at the other end the angles measured 121 and 40 respectively. 
Find the distance PQ. 

9. Find the perimeter of a regular polygon of 7 sides inscribed in 
a circle whose radius is 10 feet. Ans. 60.75 ft. 

10. The sides of a triangle are a = 10, b = 12, c = 15 respectively. 
Find the radii of the inscribed and of the circumscribed circles and 
the angles of the triangle. Ans. r = 3.23, R = 7.52, 

^ = 4 i 3o',5=53°,C=85 3o'. 



3-4] INTRODUCTION 7 

3. Inadequacy of the Graphic Method. The graphic method of 
solving triangles, though exceedingly simple and useful, is not suffi- 
ciently accurate for all purposes. For instance, in the last problem 
of Exercise 2 the results obtained by the graphic method are: 

r= 3 .2 3 , £=7.52, A = 41° 30', B=S3°, C=8 5 ° 3 o', 

while the more accurate results, obtained by a method to be de- 
scribed later, are: 

r = 3.2331, ^= 7-5 2 36, A = 4i°38 / 59 ,/ , 
B= 52° 53' 27", C= 85° 27' 34". 

The causes of the inaccuracies are manifold. First of all the 
divisions of the scale used in measuring are not indefinitely small; 
if they were the eye could not distinguish them. Besides this 
the graphic method is subject to many other unavoidable errors. 
The instruments employed in the construction of lines and angles 
are imperfect. The straight-edge is not perfectly straight, the 
divisions of the scale are not exactly equidistant. The points used 
in the construction are not true points but dots having dimensions; 
likewise the lines drawn are not true lines but pencil or pen marks 
of unequal widths. Again, neither the hand which draws the pencil 
nor the eye which guides it is perfectly steady, and so on. All 
these sources of error are unavoidable. By employing better instru- 
ments and by using greater care, these errors may be diminished, 
but they cannot be entirely eliminated. 

4. Definition of Trigonometry. There exists another method of 
solving triangles which is free from all the errors above mentioned 
and enables us to obtain results correct to any desired degree of 
accuracy. This method consists in the computation of the unknown 
parts of a triangle from the numerical values of the given parts. 
The development of this method has given rise to a separate branch 
of mathematics, called trigonometry.* Trigonometry considers the 
properties of angles and certain ratios associated with angles, and 

* The word Trigonometry comes from two Greek words, trigonon = triangle, 
and metron = measure. The method was originated in the second century B.C. 
by Hipparchus and other early Greek astronomers in their attempts to solve certain 
spherical triangles. The term trigonometry was not used until the close of the 
sixteenth century. 



8 PLANE TRIGONOMETRY [chap, i 

applies the knowledge of these properties to the solution of triangles 
and various other algebraic and geometric problems. Incidentally 
trigonometry considers also certain time-saving aids in computation 
such as logarithms, which are generally employed in the solution of 
triangles. Briefly stated, — 

Trigonometry is the science of angular magnitudes and the art of apply- 
ing the principles of this science to the solution of problems. 



CHAPTER II 
TRIGONOMETRIC FUNCTIONS OF AN ACUTE ANGLE 

5. Definition of Function. When two variables are so related that 
the value of the one depends upon the value of the other, the one is said 
to be a function of the other. 

Examples. The area of a square is a function of its side. The 
volume of a sphere is a function of its radius. The velocity of a 
falling body is a function of the time elapsed since it began to fall. 
The output of a factory is a function of the number of men employed. 

(x — i ) 
In the expression y = ) . ( , y depends upon x for its value, hence 

\X ~T~ 1/ 7 

y is a function of x. Similarly x 2 — i, # 3 + x — 3, ax -\ c, are 

functions of x. t 2 — 3 t is a function of t, etc. 

6. Definition of Reciprocal. // the product of two quantities 
equals unity, each is said to be the reciprocal of the other. 

For example, if xy = 1, x is the reciprocal of y, and y is the recipro- 
cal of x. J is the reciprocal of 2, and 2 is the reciprocal of J, for 

a b Q b -1- 

|X2 = i. In general, - and - are reciprocals since - • - = 1. .brom 

xy = 1 it follows that x = - , and y = - , that is, — 

The reciprocal of any quantity is unity divided by that quantity. 

7. The Six Trigonometric Functions of an Acute Angle. Let A 
be any acute angle, B any point on either side of the angle, and ABC 
the right triangle formed by drawing a perpendicular from B to the 
other side of the angle. Denote AC, the side adjacent to the angle 
A, by b (for base), BC, the side opposite the angle A, by a (for alti- 
tude), and the hypotenuse AB by h. 

The three sides of the right triangle form six different ratios, 
namely, 



and their reciprocals 



a 


b 


a 


V 


V 


b 


h 


h 


b 


— ) 

a 


V 


a 




Fig. 7. 




IO PLANE TRIGONOMETRY [chap, h 

a. These ratios do not depend upon the distance of the point B 
from the vertex of the angle; that is, each of the six ratios will have 
the same value for every other point B' located on either one of the 
sides of the angle. 

For if B' be any other point on AB or AB produced, or on A C or 
AC produced, and the perpendicular B'C be drawn to the other 
side, the triangle AB'C will be similar to triangle ABC and therefore 

B'C = BC a 
AB' AB~ h' 

and similarly for each of the other 

ratios. 

b. The ratios differ for different angles, 

for if the ratios were equal, the corre- 
sponding triangles would be similar (why?), and corresponding angles 
equal, which is contrary to the hypothesis that they are different. 

Since the ratios depend upon the angle for their values, they are 
functions of the angle according to the general definition of a func- 
tion given in 5. Each of these functions has received a special name. 
Referring to Fig. 7, 

a . . side opposite angle A . „ , , . , , , 

7 , that is, r^ , is called the sine of angle A ; 

h hypotenuse 

b . , . side adjacent angle A ,-,■,■, . - , 

7 , that is, ^ — , is called the cosine of angle A ; 

h hypotenuse 

a . side opposite angle A . 

- , that is, — — 7 — ~7 — 7 — 7- , is called the tangent of angle A ; 

side adjacent angle A 

h 



Fig. 8. 



the reciprocal of the sine, is called the cosecant of angle A ; 
the reciprocal of the cosine, is called the secant of angle A ; 
the reciprocal of the tangent, is called the cotangent of angle A. 



The six functions just defined are variously known as the trigo- 
nometric, circular, or goniometric functions: trigonometric, because 
they form the basis of the science of trigonometry; circular, because 
of their relations to the arc of a circle, as will be shown presently; 
goniometric, because of their use in determining angles, from gonia, 
a Greek word meaning angle. The expressions sine of angle A, 



7] TRIGONOMETRIC FUNCTIONS OF AN ACUTE ANGLE II 

cosine of angle A, etc., are abbreviated to sin A, cos A, tan A, esc A 
or cosec A , sec A , cot A . 

Besides these six functions, two others are sometimes used,* viz., 

versed sine A = i — cosine A , abbreviated to vers A ; 
coversed sine A = i — sine A, abbreviated to covers A. 

The definitions of the first six trigonometric functions must be 
thoroughly memorized. The first three are especially important and 
should be memorized in the following form : 

Given an acute angle in a right triangle, — 

The sine of the angle is the ratio of the side opposite the angle to 
the hypotenuse. 

The cosine of the angle is the ratio of the side adjacent the angle to 
the hypotenuse. 

The tangent of the angle is the ratio of the side opposite the angle 
to the adjacent side. 

The remaining three functions may be remembered most readily 
by the aid of the reciprocal relations, — 

sin A • cosec A = i, 

cos A • sec A = i, 

tan A • cot A = i, 
that is, 

cosec A = - — - » sec A = > cot A = 



sin A cos A tan A 

It will aid the memory to observe that only one o appears in each 
pair of reciprocal functions. 

It should be noticed that while a, b, and h are lines, the ratio of 
any two of them is an abstract number; that is, the trigonometric 
functions are abstract numbers. Again, the expressions sin A , cos A , 
tan A, etc., are single symbols which cannot be separated, sin has 
no meaning except as it is associated with some angle, just as the 
symbol \/ has no meaning_ except when used in connection with 
some quantity, as in va, V 4, etc. 

Example i. The sides of a right triangle are 3, 4, 5. Find all the 
trigonometric functions of the angle A opposite the side 3. 

* Especially in navigation. 



12 PLANE TRIGONOMETRY [chap, n 

Solution. The hypotenuse of the triangle equals 5. Hence, apply- 
ing the definitions, we have 

side opposite _ 
sin J± — ~, ■ — 5-. 

hypotenuse 

Similarly, cos A = f , tan A = f . 

The cotangent of A is the reciprocal of the tangent, hence 
cot A = 1 -s- f = f, similarly, sec ^4 = 1 -f- f = f , esc yl = f . 

Example 2. The legs of a right triangle are a and b; find the 
functions of the angle A opposite the side a. 

Solution. The hypotenuse h =v a 2 + 6 2 ; hence 

. . a . b b .a 

sin ^4 = 7 = ■ ? cos A = j = > tan A = - , 

h Va 2 + b 2 h Va 2 + b 2 b 



. h Va 2 + b 2 . h Va 2 + b 2 A b 

esc A = - = ? sec A = ~ = > cot A = - . 

a a 00 a 

Exercise 3 

1. A right triangle has its sides equal to 5, 12, 13. Calculate the 
six functions of the angle A opposite the side 5. 

Ans. sin A = T 5 g, cos A — \%, tan A = T 5 2, 
esc A = -VS sec A = jf , cot A = -\ 2 -. 

2. In the same triangle calculate the functions of the angle B 
opposite the side 12. 

Ans. sin B = if, cos B = T 5 3, tan B = - 1 /, 
esc B = |f, sec B = V 1 , cot B = T 5 2. 

3. By comparing the answers in problem 2 with those in problem 
1, and remembering that A + B = 90 , write down six equations, of 
which the following is the first: sin A = cos B = cos (90 — A). 

Ans. cos A = sin B = sin (90 — A), 
tan A = cot B = cot (90 — A), 
sec A = esc B — esc (90 — A), etc. 

4. Show that for any acute angle the following equations are true: 

sin (90 — A) = cos A, sec (90 — A) = esc A, 

cos (90 — A) — sin A, esc (90 — A) = sec A, 

tan (90 — A) = cot A, cot (90 — A) = tan A. 

(Suggestion. Consider A and 90 — A the two acute angles of a 

right triangle, and express the functions of each angle in terms of 

the sides of the triangle.) 



7] TRIGONOMETRIC FUNCTIONS OF AN ACUTE ANGLE 13 

5. The two legs of a right triangle are 8 and 15. Write down the 
functions of the angle A opposite the side 8; also the functions of the 
angle B adjacent the side 8. 

Ans. sin A = T 8 T , cos A = {f, tan A = T %-, 
sin B = \-% cos B = j 8 T , tan B = - 1 /-. 

6. By using the results of problem 5, show that 

sin A 41 cos A 

1 = tan A and — — 7 = cot A. 

cos A sin A 

7. One side of a right triangle is 9 and the hypotenuse is 41. 
Compute the functions of the angle A included between the hypote- 
nuse and the given side. 

Ans. sin A = ff, cos A = -£ T , tan A = %° , 
esc /x —— 4(jj sec /x ^— ~~9~) cot A. -— 4Q". 

8. Two legs of a right triangle are p 2 — q 2 , and 2 pq respectively. 
Find the sine, cosine and tangent of the angle B opposite the 
side 2 pq. 

a - t> 2 Pq -d P 2 — 9 2 -n 2 Pg 

Ans. sm B = f\ , cos B = \ % , tan B = ™ 
p 2 + q 2 p 2 + q 2 p 2 — q 2 

9. Given sin A = f , find vers A and covers A. 

Ans. vers A = I, covers A = f . 

Exercise 4 

1. Compute all the functions of 45 . 

Ans. sin 45°= cos 45°= i V 2 = 0.707, tan 45°= 1, 
esc 45°= sec 45°= V^ = 1.414, cot 45°= 1. 

(Suggestion. Construct a right triangle having an angle 45 , and 
denote each of the equal sides by a.) 

2. Compute all the functions of 30 . 

Ans. sin 30°= J, cos 30°= §^3, tan3o° = Jv / 3, 

esc 30 = 2, sec 30 = f V 3, cot 30°= V3. 
(Suggestion. If one angle of a right triangle is double the other, 
then the hypotenuse is double the shorter side. Call the shorter 
side a.) 

3. By measurement find the functions of 15 . 

Ans. sin 15°= 0.26, cos 15°= 0.97, tan 15°= 0.28, 
esc 15°= 3.87, sec 15°= 1.03, cot 15°= 3.73. 

(Suggestion. By means of a protractor, or otherwise, construct an 



14 PLANE TRIGONOMETRY [chap, ii 

angle of 15 . Draw a perpendicular to one side forming a right 
triangle. Measure the sides of the triangle, and compute the ratios.) 

4. Given sin A = f, construct and measure the angle. 

Ans. A = 45 35'. 

5. Given cos A = f, construct and measure the angle. 

Ans. A = 48 11'. 

6. Given tan A = f , construct and measure the angle. 

Ans. A = 50 12'. 

7. Show that 

sin A . cos ^4 . 

= tan ^4, - — 7 = cot A. 

cos ^4 sin yl 

A being any acute angle. 

8. Show that 

sin 2 A + cos 2 A = 1,* 
tan 2 A + 1 = sec 2 A, 
cot 2 A + 1 = esc 2 A. 
(Suggestion. Remember that a 2 + b 2 = h 2 .) 

9. Given sin A = ^, find all the other functions. 

Ans. sin A — ^5, cos ^4 = ft, tan ^4 = a?, 
esc ^ = - 2 fS sec A = If, cot A = - 2 T 4 -- 

10. Show that as an angle increases from o° toward 90 , its sine, 
tangent and secant increase, while its cosine, cotangent and cose- 
cant decrease. 

11. Show that every sine and cosine is a proper fraction, while the 
tangent and cotangent may have any value large or small. 

8. Trigonometric Functions Determined Approximately by 
Measurement. There are various ways of computing the trigono- 
metric functions of a given angle. The results of such computa- 
tions for the sines, cosines, tangents and cotangents of angles be- 
tween o° and 90 have been put together into tables, known as 
tables of natural functions. We shall learn how to use such tables 
and later how to calculate them. It is of value to the beginner to 
know how approximate values of the functions may be obtained 
graphically. 

* sin 2 A means (sin;!) 2 , tan 2 A means (tan A) 2 , etc., and generally sin 71 A = 
(sin A) n , tan n A = (tan A) n , etc., except when n = — 1. The meaning of sin -1 A, 
tan -1 A, etc., will be explained later. 



sj 



TRIGONOMETRIC FUNCTIONS OF AN ACUTE ANGLE 



A^N 



Example i. Find graphically the functions of 25 . 

Solution. By means of a protractor construct an angle MON = 
2 5 . Take OB any convenient length, say 
10 inches, and from O as a center and with 
OB as a radius describe an arc BC cutting 
ON in C. Draw CE and BA perpendicular 
to OB. By measurement we find CE = 4.23 
inches, OE = 9.06 inches, AB = 4.66 inches, 
and by construction OB = OC =10 inches. Hence 




E B 



Fig. 9. 



. 


25° 


EC 
~ 0C~ 


4-23 _ 
10 


0.423, 


esc : 


>5° = 


I 


sin. 


sin 25 




25° 


OE 

" oc = 


9.06 
10 


0.906, 


sec 


25° = 


1 


COb 


cos 25 


tan 


25° 


BA 

~ OB = 


4.66 
10 


= 0.466, 


cot 


*S°- 


I 


tan 25 



= 2.364, 



= 1. 104, 



= 2.145. 



Observe that we have used two different triangles, the triangle COE 
to obtain the sine, cosine and their reciprocals, and a second triangle 
AOB to obtain the tangent and cotangent. This was done in order 
to have in each case 10 for a divisor. If for instance we had used 
the triangle AOB only, we would have had 



sin 25 



BA 4.66 . . -4.23 . 

= -pr-z = = 0.423, instead of — -as above. 

OA 11.03 ° 10 



Exercise 5 

1. Obtain by measurement the sine, cosine and tangent of 40 . 

Arts, sin 40°= 0.643, cos 4°°= 0.766, tan 40°= 0.839. 

2. Find the sine, cosine and tangent of 35 . To avoid constructing 
the triangle and measuring the necessary lines, we may make use of 



the diagram facing page 16. sin 35 = 



AB 
OA 



Now OA=OR= 100, 



and the measure of AB = 57.4 may be read off on the vertical scale. 
mce sin 31 
Similarly, 



Hence sin 35°= ^— — 0.574 



100 



cos 35 = 



OB_ 81.9 
OA 100 



= 0.819. 



i6 



PLANE TRIGONOMETRY 



[chap, n 



To find the tangent it is better to use the triangle OPR, from which 



tan 35 = ^ 



RP 70 



100 



0.70. 



3. With the aid of Fig. 10 find each of the results given in the 
following table. 

(Suggestion. Choose your triangle so that the denominator of the 
fraction equals 100 or some other integer.) 







NATURAL 


FUNCTIONS 






Angle. 


sin 


cos 


tan 


cot 




5° 


0.087 


0.996 


0.087 


11.430 


K 


IO° 


0.174 


0.985 


0.174 


5-671 


8o° 


i5° 


O.259 


0.966 


0.268 


3-73 2 


K 


20° 


0.342 


0.940 


0.364 


2.747 


7°° ■ 


25° 


O.423 


0.906 


0.466 


2.145 


65° 


3 0° 


0.500 


0.866 


°-577 


i-73 2 


6o° 


35° 


°-574 


0.819 


0. 700 


1.428 


K 


40 


0.643 


0. 766 


0.839 


1. 192 


5o° 


45° 


0. 707 


0. 707 


1 .000 


1. 000 


45° 




cos 


sin 


cot 


tan 


Angle 



Explanation of table. For angles in the left-hand column the 
names of the functions appear on top, for angles in the right-hand 
column the names of the functions appear at the bottom. Thus 
the number 0.423, which is in the column headed "sin" and has 25 to 
the left of it, is the sine of 25 ; the same number being in the column 
which has " cos " at the bottom, and having 65 opposite it in the 
right column, is also the cosine of 65 . 

4. By use of the table, express the numbers 0.174, 0.866, 0.643, 
0.707, both as sines and cosines. 

Ans. 0.174 = sin io° = cos 8o°, 0.866 = sin 6o° = cos 30 , etc. 

5. By use of the table, express the numbers 0.364, 2.145 an d 1.000 
both as tangents and cotangents. 

6. Every number in the second and third columns is the sine of 
one angle and the cosine of another angle. What relation do you 
observe between each pair of angles? 

Every number in the fourth and fifth columns is the tangent of 
one angle and the cotangent of another. What relation exists be- 
tween each pair of angles? 



g] TRIGONOMETRIC FUNCTIONS OF AN ACUTE ANGLE 17 

7. By examining the table, verify the following statements and, 
if you can, give reasons for them. 

a. Every sine and cosine is a fraction less than unity. 

b. Every tangent of an angle less than 45 is a fraction less than 1. 

c. Every tangent of an angle greater than 45 is some number 

greater than unity. 

d. The sine and tangent of an angle increase as the angle in- 

creases. 

e. The cosine and cotangent of an angle decrease as the angle 

increases. 
/. The sine and tangent of a small angle are nearly equal. 

9. Given One of Its Functions, To Construct the Angle. In 

the last article it was shown how to find by measurement the func- 
tions of a given angle. We will now consider the converse problem, 
that is, how to construct the angle when one of its functions is given. 

Example i. To construct an angle whose tangent is f . 
Solution. Take AC 4 units in length and at C construct a per- 
pendicular CB 3 units in length. Join A and B. Then CAB is the 

required angle, for tan CAB = — ^ = f . 

The expression 

A is the angle whose tangent is f 

is written in short 
Fig. 11. 

A — tan -1 f, 
and is read in either of three ways: 

a. A is the angle whose tangent is f . 

b. A is the inverse tangent f . 

c. A is the arctangent f . 
Similarly, if 

y = sin x 

then x = sin _1 v, 

in words, if y equals the sine of x, 

then x equals the angle whose sine is y, 

or x equals the inverse sine y, 

or x equals the arcsine y. 




18 PLANE TRIGONOMETRY [chap, ii 

Corresponding meanings are given to the symbols 

y = cos -1 :r, y = csc -1 £, y = sec -1 x, y = cot -1 #. 

Example 2. To construct an angle whose sine is f . 

Solution. At a point C in a line CD of indefinite length, construct 
a perpendicular CB 4 units in length. From B as 
a center, with a radius 7 units in length, draw an 
arc cutting CD at A. Join A and B. Then the 
angle CAB is the required angle, for \y^s^\ 

- r> a r> CB 4 

sin CAB = -r-r = - . 
AB 7 

Another way of constructing sin -1 7 is to construct a semicircle 
having a diameter AB 7 units long. From B as a center, with a 
radius 4 units long, describe an arc cutting the semicircle at C. Join 
A and C. Then BAC is the required angle. (Draw the figure and 
give reasons.) 

Example 3. In the preceding example, find each of the other 
functions of the angle A. 

Solution. Two sides of the right triangle ABC being known, the 
third side is easily found. 




AC = VAB 2 - BC 2 = Vf - 4 2 = V33. 
Hence 

. AC V^ , BC 4 

C0S ^ = ZB = ~T = ' t ^ A= AC = VT3 = °- 6g6, 

secA = c^rz = ^ =i - 218 ' cot^ = -i-=-^= 1.436, 

cos a v 33 tan A 4 

csc A = — = L— 1.750. 

sin A 4 

Observe that the expressions, — 

/ " / 

sin -1 -, cos -1 — 33, tan -1 — ^, csc -1 ^, sec -1 —?^, cot -1 — 23 
7 7 V33 4 V33 4 

all represent the same angle. 

Exercise 6 

Construct an acute angle equal to A, when, — 
1. sin A = f. 2. tan A = f. 3. cos ^4 = 0.5. 

4. sec A = |. 5. tan A = 4. 6. cos ^4 = 0.45. 



10] TRIGONOMETRIC FUNCTIONS OF AN ACUTE ANGLE 1 9 

7. cos A = — • 8. sin ^4 = k. 9. tan ^4 = -. 

n k 

10. Read in three different ways the expressions, — 

A = sin _1 §, B = tan -1 3, x = esc -1 3.5, ;y = cot _1 V3. 

11. Construct and measure in degrees each of the following 
angles, — 

A = sm~ 1 i, B = cos -1 i, C=taxr l \, Z>=cot _1 J. 
Arts. ^4 = 19.5°; 5=70.5°; C=i8.5°; £=71.5°; approximately. 

12. Given sin A = f ; to find the other functions. 

v/~~ 2 — 

-4«s. cos ^4 = — 5 , tan A = "7= = § v 5, 

3 V 5 

sec A = —F = f V5. cot ^4 = J V 5 esc A = § 

vs 

13. tan 5 = V-; nn d the other functions of the angle B. 

Am. sinB =if, cos 5 = T 8 T , cot B = 1%, secB = - 1 / - - 

14. ^4 = sin - * J ; find the functions of A . 

Am. sin ^4 = ^, cos A = i v 3, tan ^4 = ^3, esc ^4 = 2, etc. 

15. Show that 

sin -1 j% = cos -1 J§ = tan -1 T 5 2. 

16. sin -1 T y = cos -1 x = tan -1 y; find x and 3/. 

^4^5. a? = if , y = T V 



17. If y = sin x, show that x = cos 'Vi — y 2 = tan * / 2 ~ 

18. Show graphically that tan -1 J + tan -1 J = 45°. 
(Suggestion. Construct tan -1 \ and tan -1 |, measure each and add 
the results.) 

10. Functions of Complementary Angles. Let ABC be any 
right triangle, C the right angle, and a, b, c the three sides. Then 
b angle A plus angle B equals 90, that is, A and B 
are complementary angles. Now 

sin A — - , and also cos B = - , 
c c 

therefore 

sin A = - = cos B = cos (90° — ^4)- 
Fig. 13. c 




20 PLANE TRIGONOMETRY [chap, ii 

Similarly 

cos A = - = sin B = sin (90 — A), 
c 

tan A = - = cot B = cot (oo° — 4), 

cot 4 = - = tan 5 = tan (90 -A), 
a 

sec A = - = esc B = esc (90 — A), 
b 

esc A = - = secB = sec (90 — A), 
a 

These results may be put in words as follows: 

The sine of an angle equals the cosine of the complement of the angle. 
The cosine of an angle equals the sine of the complement of the angle. 
The tangent of an angle equals the cotangent of the complement of the 
angle, and similarly for each of the remaining functions. 

If we arrange the six functions in three pairs, viz., sine, cosine; 
tangent, cotangent; secant, cosecant; and call either function of a 
pair the cofunction of the other, we may express the six rules just 
given by a single rule, namely: 

Any cofunction of an angle is equal to the corresponding function of 
the complement of that angle* 

By means of this rule, any function of an acute angle can be ex- 
pressed as some function of an angle less than 45 . Thus 

sin 75 = cos (90 - 75 ) = cos 15 , 
cos 8o° = sin (90 — 8o°) = sin io°, etc. 

Example. Given cot A = tan 8 A , to find one value of A . 
Solution, cot A = tan (90 — A), and from the condition of the 
problem 

cot A — tan 8 A , 

therefore tan (90 — A) — tan 8 A. 

* The term cosine was not used until the beginning of the 17th century. Be- 
fore that time the term sine of the complement (complimenti sinus) was used instead 
a contraction of which gave rise to the present name cosine. Similarly cotangent 
is a contraction of complimenti tangens and cosecant of complimenti secans. The 
abbreviations sin, cos, tan, etc., did not come into general use until the middle of 
the 1 8th century. 



it] trigonometric functions of an ACUTE ANGLE 21 



This last equation is satisfied when 

9 o°- A = SA. 

Solving A = io°. 

Note. This is not the only value that A may have. After the definitions of 
the functions have been extended to angles greater than 90 , it will be seen that 
30 , 50°, 70 , etc., are other values of A satisfying the equation cot A =tan 8 A. 

11. Functions of o°, 30 , 45 , 6o°, 90 . There are certain values 
of the angle for which the values of the functions may be easily 
determined exactly. 

a. The functions of o°. Let A be the angle formed by a fixed 
line OX and a fine OP of constant length h rotating about as a 
center. Let b be the base and a the altitude of the triangle formed 

by dropping a perpendicular from P to OX. 

As the angle increases, a increases and b de- 

x creases, and vice versa, as the angle decreases 

a decreases and b increases. As A approaches 




Fig. 14. 

o°, a approaches o and b approaches h, hence in the limit 



o o oh . o o 

sin o = — = o, cos o = - = 1, tan o = - = o. 

h h h 



b. The functions of J0°. When the angle 
A = 30 , the other acute angle of the right tri- 
angle equals 6o°; the right triangle then forms 
one-half of an equilateral triangle. Each side 
of this triangle equals h, hence a, the altitude 

of the right triangle, equals - , and the base 

2 




= y h 2 - (-J = V%h 2 =ih V s , and we have 



i h 

SU1 3 ° = V ' 



cos 30 = 

\h 



._ A/zV 



'h 



3 = W 



3, 




Fig. 16. 



tan 30 = ~~T~7= = ~ ~r = \ ^3- 

c. The functions of 45 . When A = 45 ° the 
right triangle is isosceles, that is, a and b are equal, 
and we have 

h 2 = a 2 + a 2 = 2 a 2 , a = h Vj = \h V2, so that 



22 



PLANE TRIGONOMETRY 



[chap. II 



sm 45 = 



hV] 



- = J V 2 , cos 45^ 



. hhV 



h 



h 



- 2 = *v; 



tan 45 - 



_\h^ 2 



hhV: 



- = I. 



d. T^e functions of 6o°. 6o° is the complement of 30 , hence by 
Art. 10, 

sin 6o°= cos 30 = \ V3, cos 6o°= sin 30 = J, 

, , o sin 6o° /- 

tan 60 = — v -z 

cos 60 °* 

e. The functions of 90°. As the angle A approaches 90 , a ap- 
proaches h, while b approaches o, so that in the limit 



p ■ o h oO , o h ^ ± 

sm 90 = - = 1, cos 90 = — = o, tan 90 = — = * .* 




h 



The sine, cosine, and tangent of each of the angles, 
o°, 30 , 45 , 6o°, 90 , must be remembered once for all. 
. x For convenience in memorizing, the results are collected 
in the following table. 





o° 


3°° 


^45° 


6o° 


90 


sin 
cos 
tan 



1 



1 

1^3 


IV2 

I 


^3 
1 
2 

V3 


1 

00 



V2 = I. 414 

v'i = 1 . 73 2 



A remarkaole uniformity is brought out by writing the above 
values in another form as in the table below. The student should 
verify the results for himself. 

* This needs explanation. What we have is tan A = .=- . Now as A approaches 
90 , the denominator b grows smaller and smaller, while the numerator a ap- 
proaches the constant length h. In consequence, the fraction - grows larger and 

larger. When b becomes sufficiently small, - becomes larger than any assignable 

quantity, and this is conveniently expressed by saying that as b approaches o, 7 
approaches infinity. In symbols, 



tan 90 = - = 00 . 
o 



ii) TRIGONOMETRIC FUNCTIONS OF AN ACUTE ANGLE 23 





o° 


30° 


45° 


6o° 


90 


sin 


|Vo" 


§v^ 


\yf* 


1^3 


|V4 


cos 


W4 


1^3 


\V~2 


iVx 


|V5 


tan 





i^~3 


W3 2 


l^3 3 


00 



Exercise 7 

1. Express as functions of the complementary angles, 

sin 40 , cos 50 , tan 48 , cot 22.5 , sec 6o°, esc 56 20'. 

2. Express as functions of an angle less than 45 , 

tan 75 , cos 67 30', sin 57 35', sec 73 45' 30". 

3. Find the functions of 30 by using a triangle whose hypotenuse 
is 2. 

4. Find the functions of 45 by using a triangle whose base is 1. 

5. Find the functions of 90 by using a triangle whose base re- 
mains constant and equal to 1. 

6. 90 is the complement of o°. Hence find the functions of 90 
from those of o°. 

7. Make a table containing the values of the secant, cosecant 
and cotangent of each of the angles o°, 30 , 45 , 66°, 90 . 

8. Given cos A = sin A, find one value of A. Ans. A = 45 . 

9. Given tan A = cot (45 -{-A), find one value of A. 

Ans A = 22°3o / . 
10. Given tan 2 A = cot 3 A, find one value of A. 

Ans. A = 18 . 
n. Given sin (45 + A) = cos (30 + A), find one value of A. 

Ans. A = 7°3o r . 

12. Given sin 2 A = cos (45 — A), find one value of A. 

Ans. A = 45 . 

13. Given cot A = tan (n — 1) A, find one value of A. 



Ans. A = $2- 
n 



14. Prove that vers A = covers (90 — A), 
and that covers A — vers (90 — A). 



24 



PLANE TRIGONOMETRY 



[chap, n 



15. It is shown in geometry that if the radius of a circle is divided 
into extreme and mean ratio, the greater segment will be equal to the 
chord subtending an arc of 36 ; that is, if in Fig. 18 
AO is divided at B so that AO : OB = OB : BA, 
then the chord AC, taken equal to OB, subtends 
an angle CO A equal to 3 6°. 

Put AC = OB = x, and AO = r, then CD A is 
a right triangle whose short side is x, whose 
hypotenuse is 2 r, and the angle CDA = iS°. 

Hence sin 18 = — , where x is the positive root of 
2 r 

the equation 

r : x = x : r — x. 

Solving, we find the positive value oi x, x = r { — 1+2^5)- 




Hence 



sin i< 



= JL = ri-h + hVj) = . ( _ j +^). 
2 r 2Y 

v IP— 2 \J$ 

V5 



Show now that cos 18 = | V 10 -f- 2V5, sec 18 = 



tani8°=— 25 



V 2C — IQ V5 



, cot 18 = V 5 + 2 V s . 



12. Fundamental Relations. In any right triangle (Fig. 19) 
1 a 2 + b 2 = h 2 . 

a By dividing this equation first by h 2 , then by 
b 2 , and finally by a 2 , we obtain in turn 




Fig. 19. 



-]+(-)= i, or sin 2 ^4 + cos 2 ^l = 1, 
h) \h) 

(-Y+ 1 = (-), or tan 2 .4 + 1 = sec 2 ^., 

1 + (~J= (-\\ or 1 + cot 2 .4 = csc 2 ^. 



W 

(2) 
(3) 



(1), (2), and (3) are called the square relations of the trigonometric 
functions. These together with the three reciprocal relations 

smA • csc^L = 1, (4) 

cos ^4 • sec A = 1, (5) 

tan^t • cot ^4 = 1, (6) 



13] TRIGONOMETRIC FUNCTIONS OF AN ACUTE ANGLE 25 

constitute the six fundamental relations of the trigonometric func- 
tions. To these, two others are usually added, viz., 

tlmA = **A, cotA = c -™^. ( 7 ) 

cos A sin A 

Proof. 

a 

sin A ha, , ■, cos A 1 , 

= - = - = tan A , and — = = cot A . 

cos Abb sin A tan A 

h 

13. To express each of the Functions in Terms of a Given 
One. 

a. Analytic method. Example i. To express each of the functions 
in terms of the sine. 

(By (1), Art. 12) 
(By (7), Art. 12) 

(By (5), Art. 12) 

(By (4), Art. 12) 

(By (7), Art. 12) 



cos A = Vi — 

, . sin A 

tan A = 

cos A 


sin 2 A 

sin A 


Vi — sin 2 A 
1 


cos A 


Vi — sin 2 A 


sin -4 




1 


v'i — sin 2 A 



tan A sin A 

Example 2. To express each of the functions in terms of the 
tangent. 



sec 



A = Vi + tan 2 ,4 (By (2), Art. 12) 



cot ,4 = —5— (By (6), Art. 12) 

tan A 

cos A = — ?— = / \ (By (5), Art. 12) 

sec A v 1 + tan 2 A 

tSLTlA . N 

sin ^4 = cos ^4 • tan A = , = (By (7), Art. 12) 

V 1 + tan 2 A 



csc A = -±- = Vl + ta " M (By (4), Art. 1 2) 

sin ^4 tan A 



26 



PLANE TRIGONOMETRY 



[chap. II 



b. Geometric method. Example i. To ex- 
press each of the functions in terms of the 
sine. 

In the right triangle ABC (Fig. 20), 

BC 




Fig. 20. 



sin A = — - ; hence if AB is chosen for the unit of measure, 
AB 

sin A = — = BC, and AC = VAB 2 - BC 2 = Vi - sin 2 4. 



Then from the definitions of the trigonometric functions 

cos.4 = — = Vi - sin 2 4, tan 4 =^= . = 

AB AC Vi-sin 2 ^ 



, etc. 



Example 2. To express each of the functions in terms of the 
tangent. 

B In Fig. 21, 

^ tan A = — ; hence if A C is chosen for the unit of measure, 
S AC 




A = — =BC, and AB = V AC 2 + BC 2 = Vi + tan 2 X 



tan 

Fig. 21. Then by definition 
BC tan 4 



sin A = 



j cos ^4 = 



AC 



AB Vi + tan 2 ^ AB Vi + tanM 

Exercise 8 

* 

1. From (1) show that 



j etc. 



sin A = V 1 — cos 2 ^4, cos ^4 = V 1 — sin 2 ^4, 
(1 — sin ^4) (1 + sin ^4) = cos 2 ^4, 

cos .4 1 + sin ^4 



(1 — cos A) (1 + cos A) = sin 2 A, 
sin ^4 



1 — sin A 
1 + cos A 



cos A 



sin ^4 



1 — cos A 
2. From (2) show that 

sec A = Vi + tan 2 ^4, tan A = Vsec 2 A — 1, 
(sec A — tan A) (sec ^4 -f- tan ^4) = 1. 



i4l TRIGONOMETRIC FUNCTIONS OF AN ACUTE ANGLE 27 
3. Show that 



esc A = Vi 4- cot 2 A, cot A = VcscM — 1, 
(esc A — cot^4) (esc A + cot A) = 1. 

4. From (7) show that 

cos A • tan A = sin A , sin A • cot A = cos A , 
sin A • sec A = tan ^4 , cos A • esc A = cot ^4 . 

5. Express in words the relations given by formulas (1) to (7). 

6. Use the analytic method to express the tangent in terms of 
the cosecant; in terms of the cosine. 

7. Use the geometric method to express each the sine and the 
cosine in terms of the tangent. 

In the following exercises compare your results with those given in 
the table on page 35. 

8. Express each of the functions in terms of the cosine. 

9. Express each of the functions in terms of the cosecant. 

10. Express each of the functions in terms of the secant. 

11. Use sin 30 = \ to find each of the remaining functions of 30 . 

12. sin 15 = \ v 2 — V3; find the other functions of 15 . 



Ans. cbsi5°=i V2+V3, tan 15°= 2 —V3, sec 15°= 2 V 2 — ^3, 

CSC15 = 2 V2 + V3, cot 15 = 2 + V3. 
13. K 



= i/ (j ~ h) {S ~ C) , where s = a -±A+J 
V be 2 



sin x = . 

DC 



prove that cos x = \J — ■ and tan x = y i x • 

V be * s (s — a) 

14. Reduction of Trigonometric Expressions to their Simplest 
Form. Like algebraic expressions, expressions involving trigono- 
metric functions may frequently be reduced to a' simpler form. As a 
rule the reduction is most easily effected by expressing each of the 
functions which occur in the expression in terms of the sine and 
cosine and by reducing the resulting expression like any algebraic 
expression, treating the sine and cosine as two separate quantities. 
In the end the result may again be expressed in terms of whatever 
function or functions give the result the simplest form. Of course 
one might express everything in terms of a single function, say the 



28 PLANE TRIGONOMETRY [chap, n 

sine, but that would introduce radicals for every function except the 
cosecant. By using both the sine and cosine radicals are avoided, 
unless, of course, the expression involves radicals to begin with. 

Example i. Reduce the expression sin A + cot A cos A. 

Solution. Substituting for cot A its value in terms of the sine 
and cosine, we have 

. A , cos A . 

sin A + — cos A . 

sin A 

Reducing to a common denominator, 

sin A sin A + cos A cos A __ sin 2 A -\- cos 2 A 
sin A sin A 

Now by (i), Art. 12, sin 2 A + cos 2 A = 1 ; hence finally, 

sin A + cot A cos A = = esc A . 

sin A 

Example 2. Reduce the expression 

sin 2 A tan A + cos 2 A cot A + 2 sin A cos A . 

Solution. Substituting for tan A and cot A their values in terms 
of the sine and cosine, (7), Art. 12, we have 

• n 4 Sill J\. 1 9 A COS J\. \ • A A 

sur A (- cos 2 ^4 — 1- 2 sin ^4 cos ^4 

cos A sin ^4 

_ sin 4 A + «cos 4 A -\- 2 sin 2 ^4 cos 2 ^4 _ _ (sin 2 A -f- cosM) 2 
sin ^4 cos v4 sin ^4 cos A 

= esc ^4 sec A. 



sin ^4 cos ^4 



Example 3. Reduce the expression 

(cos x cos y — sin x sin y) 2 + (sin x cos y -f- cos x sin y) 2 . 

Solution. Squaring the expressions enclosed in parentheses, 

(cos x cos y — sin x sin y) 2 = cos 2 x cos 2 y — 2 cos x cos y sin # sin y 

+ sin 2 x sin 2 y, 

(sin x cos y + cos # sin y) 2 = sin 2 x cos y + 2 sin x cos y cos x sin y 

+ cos 2 x sin 2 y. 

Adding the right-hand members of the last two expressions, we have 

cos 2 y (cos 2 x + sin 2 x) + sin 2 y (sin 2 # + cos 2 x) 
= cos 2 y + sin 2 y = 1. 



14] TRIGONOMETRIC FUNCTIONS OF AN ACUTE ANGLE 29 

Complicated expressions, involving a single angle, may be most 
easily reduced by putting for each function its value in terms of the 
sides a, b, h of a right triangle. The resulting expression may then 
be reduced like any other algebraic expression. Of course, the re- 
lation h 2 = a 2 + b 2 may be made use of whenever an advantage is 
gained by it. 

Example 4. Reduce the expression 

sin 2 A tan A + cos 2 A cot A + 2 sin A cos A . 

Solution. Putting 

a & a b a a 4. a b 

smA = - , cos A = - , tan A = - , cot A = - , 

h h b a 

we have 

a?a,Pb J _ 2 ab == a 4 +b*+2 a 2 b 2 = (a 2 + b 2 ) 2 = a 2 + 6 2 = a , & 
/f 2 & /z 2 a hh abh 2 abh 2 ab b a 

— tan A + cot A . 

The preceding methods are perfectly general, but frequently it is of 
advantage to use other expedients. Any one of the seven relations in 
Art. 12 may be employed in the reduction. Sometimes the denomi- 
nator may be simplified or removed by multiplying both terms of 
the fraction by a binominal factor like 1 — cos A, 1 + sin A, sec A 
— tan A, esc A + cot A ; in short, as in algebra, the form of the ex- 
pression may be changed by any operation which does not change 
the value of the result. Radical expressions should be avoided 
whenever possible. 

Example 5. Reduce the expression 

.,',,2 



Solution. 



Example 6. Reduce 



sur x 
1 — cos x 

sin 2 a; 1 -j- cos x _ sin 2 x (1 -f- cos x ) 
1 — cos x 1 + cos x 1 — cos 2 x 

= 1 + cos x, since 1 — cos 2 x = sin 2 x. 

COS0* 



sec 8 + tan 6 



* Greek letters are frequently used to represent angles. For the benefit of 
those students who do not already know the Greek letters and their names, the 
Greek alphabet has been printed in the front part of this book. The Greek letters 
are written as they are printed. 



30 PLANE TRIGONOMETRY [chap, ii 

Solution. Multiplying both terms of the fraction by sec 6 — tan 0, 
we have 

cosfl _. cosfl t (sec 6— tan d) _ cos 6 (sec 6 — tan 6) 

sec 6 + tan d ' (sec (9 + tan d) (sec 0- tan 6) sec 2 — tan 2 6 

= cos (sec 6 — tan 0), since sec 2 6 — tan 2 = i 
= i — sin 6, since cos • sec 6 = i , cos • tan 6 = sin 0. 

Exercise g 

Reduce the following expressions, — 
cos A i 



sin A cot 2 A 



Am. tan A 



sin ^4 i cos A A 

2. -H • Ans. i. 

esc ^4 sec ^4 

3. tan 2 x esc 2 x — 1. • Ans. tan 2 x. 

cos -v sin y A . . 

4. ■ * - — • Ans. sin y + cos y. 

1 — tan y cot 3/ — 1 

5. (tan 6 + cot 6) sin cos 0. Ans. 1. 

6. cos x tan * + sin x cot x. Ans. sin * + cos x. 

7. sec a — tan a sin a. Ans. cos a. 

8. cot 6 -\ . Ans. esc 8. 

1 + cos 6 

9. sin 4 B + cos 4 B + 2 cos 2 5 sin 2 5. ^4ws. 1. 

cos 2 x A . 

10. : — • ,4*w. i + sin x. 

1 — sin x 

11. sin ^4 (sec ^4 + esc ^4) — cos A {sec A — esc ^4). 

m Ans. sec A esc ^4. 

12. cot/3 — sec /3 esc /3 (1 — 2 sin 2 /3). Ans. tan/3. 

13. (1 + sin A) {sec A — tan A). Ans. cos ^4. 

tan A + tan B A . . , „ 

14. ! Ans. tan ^4 tan B. 

cot .4 + cot B 

15. (sin x + cos x) 2 + (sin x — cos x) 2 . Ans. 2. 

16. 2 (sin 6 + cos 6 d) — 3 (sin 4 (9 + cos 4 0) + 1. Ans. o. 

17. 2 vers ^4 — vers 2 A . Ans. sin 2 ^4. 



15] TRIGONOMETRIC FUNCTIONS OF AN ACUTE ANGLE 31 

cos A cot A — sin A tan A A , . . . 

18. • Ans. 1 + sin .4 cos .4 . 

esc A — sec A 

sin B , 1 + cos B a t> 

19. 1 — • Ans. 2 esc B. 

1 -f- cos B sin B 

20. csc 4 # (1 — cos 4 #) — 2 cot 2 #. Ans. 1. 

21. (cos # cos y + sin # sin y) 2 + (sin* cosy— cos # sin y) 2 . Ans. 1. 

22. (# cos a: — y sin a) 2 + (# sin a + 3; cos a) 2 . Ans. x 2 + y 2 . 

sec 2 ^4 sin 2 A — cscM + cscM cos z 4 ,. . 9 . 

23. 7—— -— -— Ans. sin 2 .4. 

sec 2 A sni 2 A — esc 2 ^4 cos 2 A 



4 A — sin A A a 4. a 

24. I/ • Ans. sec ^4 — tan A . 

▼ 1 + sin ^4 

15. Trigonometric Identities. Equations which express gen- 
eral relations, that is, equations which remain true no matter what 
values be given to the quantities which are considered variable, are 
called identities, (x + i) 2 = x 2 -\- 2 x + 1 is an identity, because it is 
true no matter what value be given to x. x 2 — 5 x + 6 = o is not an 
identity, since it is not true unless x has the value 2 or 3. Similarly, 

sin 2 x + cos 2 x — 1 and tan A = r are identities, for they ex- 
cos^ 

press general relations, relations which hold true no matter what x 
or A may be. On the other hand, sin 2 x — \ sin x = — \ is not an 
identity, for it is true only when sin x = \, that is, when x = sin -1 J. 
To distinguish equations which are not identities from those which 
are, the former are sometimes called equations of condition, since 
they express conditions to which the variables are restricted and 
not general relations as do identities. 

The fundamental trigonometric identities are given in Art. 12. 
All other trigonometric identities may be derived from these by 
properly combining them. To prove a given identity it is sufficient 
to reduce both sides to the same form. If no shorter way suggests 
itself, this may always be accomplished by reducing each side to its 
simplest form. 

Example i. Prove the identity 

(1 — tan A) (1 — cot A) + sec A esc A = 2. 
Solution. Putting 

. A sin A r A cos A A 1 A 1 

tan A = j cot A == > sec A = > esc A = — — - > 

cos^l sini cos^. sini 



32 PLANE TRIGONOMETRY [chap, n 

the left-hand member becomes 

sin A \ / cos A , , 



cos^4/\ sin A) cos A sin A 

__ (cos A — sin A) (sin ^4 — cos ^4) -+- i 
cos A sin A 

— 2 sm ^ cos A ~ s * n2 4 ~ cos2 -^ ~f~ i 

cos ^4 sin ^4 

_ 2 sin ^4 cos A 
cos A sin A 

n, ft Tj Tj 

If we had put tan A = - , cot A = - , sec A = - , esc A = - , the re- 

duction would have been as follows, 

d -Vi &N i 1 ^- ( 5 ~ a )^~ &) + # 

\ b/\ a) ba ab 

_ 2 ab — a? — b 2 -\- h 2 2 ab 



ab ab 



= 2. 



Example 2. Show that 

sin x tan 2 x + esc x sec 2 x = 2 tan x sec x + esc x — sin x. 

Solution. Reducing each member of the equation separately, we 

have 

, 9 , « sin 2 x 1 1 1 sin 4 x + 1 
sm x tarn x + esc x sec z x = sm x 1 — : — ■ — — ■ = — — - . 

cos 2 x smxcos 2 x sin x cos 2 x 
Likewise the second member becomes 

. . 2 sin x 1 1 1 

2 tan x sec x + esc x — sin x = ■ • ■ ■ + — — ■ — sm x 

cos x cos x sin % 

2 sin 2 x -{- cos 2 x — sin 2 x cos 2 x 
sin x cos 2 x 

2 sin 2 x + (1 — sin 2 x) cos 2 x 
sin x cos 2 x 

2 sin 2 x -j~ (1 — sin 2 x) (1 — sin 2 x) 
sin x cos 2 x 

— si 1 * 4 g ~f- 1 
sin x cos 2 x 



15] TRIGONOMETRIC FUNCTIONS OF AN ACUTE ANGLE 33 

Example 3. Is sec 2 A esc 2 A = tan 2 A + cot 2 A + 2 ? 

Solution. 

sec 2 4 esc 2 4 = (1 + tan 2 A) (1 + cot 2 4) 

= 1 + tan 2 4 + cot 2 A + tan 2 4 cot 2 A 

= 1 + tan 2 ^4 + cot 2 .4 + 1, since tan .4 cot A = 1. 

= tan 2 .4 + cot 2 ^4 + 2. 

Hence the identity is true. 

Exercise 10 
Prove the following identities: 
1. sin 2 A sec 2 A = sec 2 .4 — 1. 
. 2 . (sin B + cos B) 2 = 1 + 2 sin B cos B. 

3. sin 4 x — cos 4 x = sin 2 # — cos 2 x = 1 — 2 cos 2 x = 2 sin 2 # — 1. 

4. (sin 2 — cos 2 6) 2 = 1 — 4 sin 2 cos 2 0. 

cot ^4 cos ^4 _ cot^4 — cos A 
cot A + cos ^4 cot A cos ^4 

6. sin 3 x + cos 3 # = (sin x + cos x) (1 — sin # cos %). 

7. (sin ^4 + cos A) (tan ^4 + cot A) = sec A + esc A. 

R sec + esc 6 _ tan + 1 __ 1 + cot d 

sec 6 — esc tan 6 — 1 1 — cot 

: 9. sec 4 ^4 + tan 4 A = 1 + 2 sec 2 y4 tan 2 yl . 

10. tan a + tan /3 = tan a tan jS (cot a + cot 0) . 

11. (1 + sin x + cos x) 2 = 2 (1 + sin x ) (1 + cos x). 

tan ^4 + sec ^4 — 1 , , , , 

12. ! = tanyl + sec A. 

tan ^4 — sec A + 1 

13. cos 4 x — sin 4 x = cos 2 # (1 — tan x) (1 + tan x). 

14. cos 2 £ + sin 2 x sin 2 y + sin 2 x cos 2 >> = 1. 

15. (cos a cos /3+sin a sin /3 cos y) 2 + (sin a cos j8— cos a sin /3 cos 7) 2 

= 1 — sin 2 /3 sin 2 7. 



34 



PLANE TRIGONOMETRY 



[chap. II 



^ 




^ 




^ 


^ 


o 
o 


^ 


o 


^ 


o 
o 


o 


+ 


o 


H 
+ 


o 
u 


+ 


+ 


H 




M 




M 


M 


> 




V 




> 


V 



M 






M 




1 


^ 


^ 


1 


^ 


OJ 


CJ M 


o 




o 


CJ 


(U 


u 


cj 


<u 


<D 


w 


WD 


<d 


m 


> 






> M 





> 



> 





H 






^ 


1 


^ 




cj 




(J 




m 


( ) 


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(J 


m 
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> 


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o 
in 
u 

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M 


u 


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o 


u 


m 




cj 




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> 



a 
+ 

H 



cj 

+ 



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PJ 








a 




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c3 


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sn 




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+ 


rt 

-i-> 


+ 




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cfl 






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CHAPTER III 
SOLUTION OF RIGHT TRIANGLES BY NATURAL FUNCTIONS 

16. Tables of Natural Functions. In the preceding chapter we 
computed the functions of 30 , 45 and 6o°. Later we shall learn 
how to compute the functions of any given angle. Now we cannot 
remember the values of the functions for all angles, and the com- 
putation by which the values are determined is too laborious to be , 
repeated each time the value of a particular function is needed. 
For this reason the values of the functions for every degree, minute 
and second from o° to 90 have been computed once for all and the 
results tabulated in tables, known as tables of natural functions. 
Usually such tables contain the sines, cosines, tangents and cotan- 
gents of angles differing by 1 minute; in some tables the angles 
differ by 10 seconds, and in still others by only 1 second. From such 
tables, the value of the sine, cosine, tangent and cotangent may be 
found whenever needed, and conversely, when the value of a func- 
tion is known, the corresponding angle may be found by use of the 
tables. Secants and cosecants are not given directly by the tables, 
but may be found indirectly from the cosines and sines respectively,, 

since sec A = , and esc A = . 

cos A sin A 

17. To Find the Natural Functions of an Angle Less than 90°. 

a. When the angle is less than 45 , the degrees are found at the 
head of the column and the minutes in the left-hand column. The 
number, in the same horizontal line as the minutes and in the same 
column as the degrees, is that function of the angle whose name 
appears at the head of the column. 

Example i. To find the sine of 2 6° 31'. 

Solution. We find the column in the table of natural sines and 
cosines (specimen page, p. 40) which is headed by 26 and follow 
down the column marked sine on top until we come to the number 
4465 which is in the line beginning with 31'. The number 4465 
considered a decimal is the required sine, that is, 

sin 26°3i / = 0.4465. 

35 



36 PLANE TRIGONOMETRY [chap, in 

b. When the angle is greater than 45 , the degrees are found at the 
foot of the column and the minutes in the right-hand column. The 
number in the same horizontal line as the minutes in the right-hand 
column and in the same column as the degrees at the bottom, is that 
function of the angle whose name appears at the foot of the column. 

Example 2. To find cos 63 ° 29'. 

Solution. We find that column in the table of natural sines and 
cosines (see specimen page) which has 63 ° at its foot, and follow up 
the column marked cos at the bottom until we come to the number 
4465 which is in the horizontal line ending with 29/. The number 
.4465 considered a decimal is the required cosine, that is, 

cos 63 29' = 0.4465. 

c. When the angle consists of degrees, minutes and seconds, the 
result obtained for the given degrees and minutes must be corrected 
for the additional seconds. 

Example 3. To find sin 28°46' 36". 

Solution. 28 46' 36" = 28 46 .6'. 

From the table we find 

sin 28°46 /= 0.4812 

sin 28°47'= 0-4815 

difference for i' = 0.0003 

The angle 2 8° 46. 6' whose sine we seek is A the way between the two 
angles 28 46' and 28 47', hence we take for its sine the sine of 28 46' 
increased by ^0 of the difference between sin 28 46' and sin 28 47'. 
T e o of 0.0003 = 0.00018, or 0.0002 if we carry 4 decimal places only. 

Hence 

sin 28°46 / 36' / = 0.4812 + 0.0002 = 0.4814. 

Example 4. To find cos 6i° 13' 24". 
Solution. 6i° 13' 24" = 6i° 13.4'. 
From the table we find 

cos6i° 13' = 0.4815 

cos6i° 14' = 0.4812 

difference for 1' = 0.0003 

t*o of 0.0003 = 0.00012, or 0.0001 if we carry four places only, and since 
the cosine of 6i° 13.4' must be between cos 6i° 13' and cos 6i° 14', 

cos 6i° 13' 24" = 0.4815 — 0.0001 = 0.4814. 



17] SOLUTION OF RIGHT TRIANGLES BY NAT. FUNCTIONS 17 



^ * 



Observe that in Example 3 the difference was added to the sine of the 
smaller angle, while in Example 4 the difference was subtracted from 
the cosine of the smaller angle. This is because as the angle in- 
creases the sine increases while the cosine decreases. For the same 
reason the difference must be added to the tangent and subtracted from 
the cotangent of the smaller angle. 

The tangent or cotangent of an angle is found from the table of 
natural tangents and cotangents in exactly the same way that the 
sine or cosine is found from the table of natural sines and cosines. 

Example 5. To find tan 63 ° 16' $2" . 
Solution. 63 16' 32" = 63 16.5^. 
From the table of natural tangents and cotangents (see specimen 

page, p. 41) we find 

tan63° 16' = 1.9854 
tan63°i7 / = 1.9868 

difference for i' = 0.0014 

5^ times 0.0014 = 0.000747, or 0.0007 to four places. Hence 
tan 63 16' 32" = 1.9854 + 0.0007 = 1.9861. 

Example 6. To find cot 26 43' 28". 
Solution. 26 43' 28" = 26 43.4!'. 
From the table (specimen page) 

cot26°43 r = 1.9868 
cot 26°44 r = 1.9854 

difference for i' = 0.0014 
4§ of 0.0014 = 0.00065 or 0.0007 to four places. Hence 
cot 26°43' 28" = 1.9868 — 0.0007 = 1-9861. 

The process of finding the value of a function of an angle inter- 
mediate to two consecutive angles whose functions are given directly 
in the table, is called interpolation. Thus in Example 4, cot 36°43' 
and cot 3 6° 44' are given directly in the table, while cot 3 6° 43' 28", 
which is not given in the table, was found by interpolation. 

In interpolating we assumed that the increase or decrease of the 
function is proportional to the increase of the angle, that is, we assumed 
what is known as the principle of proportional parts. Briefly stated 
it is this, — 




$8 PLANE TRIGONOMETRY [chap, ni 

For small changes in the angle the change in the function of an angle 
is nearly * proportional to the change in the angle. 

Example 7. One angle of a right triangle is 25 48.5' and the 
hypotenuse is 235.0. Find the remaining sides of the triangle. 

Solution. In the adjacent figure, let A represent j$^, 

the given angle and c the hypotenuse. 

To find a we have, ^ffff 48 b 

A 

- = sin A , or a = c sin A. Flg ' 22 
c 

c = 235, and from the table we have sin A = 0.4354, 

hence a = 235 X 0.4354 = 102.3. 

To find b, we have, 

- = cos A , or b = c cos A , 
c 

c = 235, and from the table we have cos A = 0.9003, 

hence, b = 235 X 0.9003 = 21 1.6. 

To check our results, we use the relation - = tan A , that is, if our 

b 

results are correct, the quotient of a by b must agree with the value 
of tan A as found from the table. 

a 102 "? • • • • 

- = — = 0.4835, while tan A as given in the table is 0.4836. 

b 211. 6 

The difference of 1 in the last decimal place arises from the neglected 
parts of the decimals. If we had carried out the work to five signifi- 
cant figures instead of four, the quotient of a divided by b would 
have been 0.4836. 

Exercise ii 

All the functions called for in this exercise are found on the speci- 
men pages of natural functions on pp. 40, 41. 

1. Find sin 26 , cos 27°30 / , tan 25°45 r , cot 29°59 / . 

Ans. 0.4384, 0.8870, 0.4823, 1.7332. 

2. Find cos 64 , sin 62°3o', cot 64 15', tan 6o°oi'. 

Ans. 0.4384, 0.8870, 0.4823, 1.7332. 

* We say nearly, for no exact proportion exists, no matter how small the 
change in the angle. All we can say is, that when the change in the angle is 
small, the principle of proportional parts gives results which in most cases are suffi- 
ciently exact for practical purposes. 



17] SOLUTION OF RIGHT TRIANGLES BY NAT. FUNCTIONS 39 











S 


PECIME 


tf PAGE 












1 


25° 


26° 


27° 


28° 


29° 


' 


sin 


cos 


sin 


cos 


sin 


cos 


sin 


cos 


sin 


cos 





4226 


9063 


4384 


8988 


4540 


8910 


4695 


8829 


4848 


8746 


60 


I 


4229 


9062 


4386 


8987 


4542 


8909 


4697 


8828 


4851 


8745 


59 


2 


4231 


9061 


4389 


8985 


4545 


8907 


4700 


8827 


4853 


8743 


58 


3 


4234 


9059 


4392 


8984 


4548 


8906 


4702 


8825 


4856 


8742 


57 


4 


4237 


9058 


4394 


8983 


45SO 


8905 


4705 


8824 


4858 


8741 


56 


5 


4239 


9057 


4397 


8982 


4553 


8903 


4708 


8823 


4861 


8739 


55 


6 


4242 


9056 


4399 


8980 


4555 


8902 


4710 


8821 


4863 


8738 


54 


7 


4245 


9054 


4402 


8979 


4558 


8901 


47i3 


8820 


4866 


8736 


53 


8 


4247 


9053 


4405 


8978 


456i 


8899 


4715 


8819 


4868 


8735 


52 


9 


4250 


9052 


4407 


8976 


4563 


8898 


4718 


8817 


4871 


8733 


5i 


10 


4253 


9051 


4410 


8975 


4566 


8897 


4720 


8816 


4874 


8732 


50 


11 


4255 


9050 


4412 


8974 


4568 


8895 


4723 


8814 


4876 


8731 


49 


12 


4258 


9048 


4415 


8973 


457i 


8894 


4726 


8813 


4879 


8729 


48 


13 


4260 


9047 


4418 


8971 


4574 


8893 


4728 


8812 


4881 


8728 


47 


14 


4263 


9046 


4420 


8970 


4576 


8892 


473i 


8810 


4884 


8726 


46 


15 


4266 


9045 


4423 


8969 


4579 


8890 


4733 


8809 


4886 


8725 


45 


16 


4268 


9043 


442 5 


8967 


458i 


8889 


4736 


8808 


4889 


8724 


44 


17 


4271 


9042 


4428 


8966 


4584 


8888 


4738 


8806 


4891 


8722 


43 


18 


4274 


9041 


4431 


8965 


4586 


8886 


4741 


8805 


4894 


8721 


42 


19 


4276 


9040 


4433 


8964 


4589 


8885 


4743 


8803 


4896 


8719 


4i 


20 


4279 


9038 


4436 


8962 


4592 


8884 


4746 


8802 


4899 


8718 


40 


21 


4281 


9037 


4439 


8961 


4594 


8882 


4749 


8801 


4901 


8716 


39 


22 


4284 


9036 


4441 


8960 


4597 


8881 


4751 


8799 


4904 


8715 


38 


23 


4287 


9035 


4444 


8958 


4599 


8879 


4754 


8798 


4907 


8714 


37 


24 


4289 


9033 


4446 


8957 


4602 


8878 


4756 


8796 


4909 


8712 


36 


25 


4292 


9032 


4449 


8956 


4605 


8877 


4759 


8795 


4912 


8711 


35 


26 


4295 


9031 


4452 


8955 


4607 


8875 


476i 


8794 


4914 


8709 


34 


27 


4297 


9030 


4454 


8953 


4610 


8874 


4764 


8792 


4917 


8708 


33 


28 


4300 


9028 


4457 


8952 


4612 


8873 


4766 


8791 


4919 


8706 


32 


29 


4302 


9027 


4459 


8951 


4615 


8871 


4769 


8790 


4922 


8705 


31 


30 


4305 


9026 


4462 


8949 


4617 


8870 


4772 


8788 


4924 


8704 


30 


31 


4308 


9025 


4465 


8948 


4620 


8869 


4774 


8787 


4927 


8702 


29 


32 


43io 


9023 


4467 


8947 


4623 


8867 


4777 


8785 


4929 


8701 


28 


33 


4313 


9022 


4470 


8945 


4625 


8866 


4779 


8784 


4932 


8699 


27 


34 


43i6 


9021 


4472 


8944 


4628 


8865 


4782 


8783 


4934 


8698 


26 


35 


43i8 


9020 


4475 


8943 


4630 


8863 


4784 


8781 


4937 


8696 


25 


36 


4321 


9018 


4478 


8942 


4633 


8862 


4787 


8780 


4939 


8695 


24 


37 


4323 


9017 


4480 


8940 


4636 


8861 


4789 


8778 


4942 


8694 


23 


38 


4326 


9016 


4483 


8939 


4638 


8859 


4792 


8777 


4944 


8692 


22 


39 


4329 


9015 


4485 


8938 


4641 


8858 


4795 


8776 


4947 


8691 


21 


40 


4331 


9013 


4488 


8936 


4643 


8857 


4797 


8774 


495o 


8689 


20 


41 


4334 


9012 


4491 


8935 


4646 


8855 


4800 


8773 


4952 


8688 


19 


42 


4337 


901 1 


4493 


8934 


4648 


8854 


4802 


8771 


4955 


8686 


18 


43 


4339 


9010 


4496 


8932 


4651 


8853 


4805 


8770 


4957 


8685 


17 


44 


4342 


9008 


4498 


8931 


4654 


8851 


4857 


8769 


4960 


8683 


16 


45 


4344 


9007 


45oi 


8930 


4656 


8850 


4810 


8767 


4962 


8682 


15 


46 


4347 


9006 


4504 


8928 


4659 


8849 


4812 


8766 


4965 


8681 


14 


47 


435o 


9004 


45o6 


8927 


4661 


8847 


4815 


8764 


4967 


8679 


13 


48 


4352 


9003 


4509 


8926 


4664 


8846 


4818 


8763 


4970 


8678 


12 


49 


4355 


9002 


45" 


8925 


4666 


8844 


4820 


8762 


4972 


8676 


11 


50 


4358 


9001 


4514 


8923 


4669 


8843 


4823 


8760 


4975 


8675 


10 


51 


4360 


8999 


4517 


8922 


4672 


8842 


4825 


8759 


4977 


8673 


9 


52 


4363 


8998 


4519 


8921 


4674 


8840 


4828 


8757 


4980 


8672 


8 


53 


4365 


8997 


4522 


8919 


4677 


8839 


4830 


8756 


4982 


8670 


7 


54 


4368 


8996 


4524 


8918 


4679 


8838 


4833 


8755 


4985 


8669 


6 


55 


4371 


8994 


4527 


8917 


4682 


8836 


4835 


8753 


4987 


8668 


5 


56 


4373 


8993 


453o 


8915 


4684 


8835 


4838 


8752 


4990 


8666 


4 


57 


4376 


8992 


4532 


8914 


4687 


8834 


4840 


8750 


4992 


8665 


3 


58 


4378 


8990 


4535 


8913 


4690 


8832 


4843 


8749 


4995 


8663 


2 


59 


438i 


8989 


4537 


891 1 


4692 


8831 


4846 


8748 


4997 


8662 


I 


60 


4384 


8988 


454° 


8910 


4695 


8829 


4848 


8746 


5000 


8660 





/ 


COS 


sin 


cos 


sin 


cos 


sin 


cos 


sin • 


cos 


sin 


/ 


64° 


63° 


62° 


61° 


60° 



4° 



PLANE TRIGONOMETRY 



[chap. Ill 



SPECIMEN PAGE 



' 


25° 


26° 


27° 


28° 


29° 


t 


tan 


cot 


tan cot 


tan cot 


tan 


cot 


tan 


cot 







4663 


2 . 1445 


4877 2.0503 


5095 1.9626 


5317 


1.8807 


5543 


1 . 8040 


60 


I 


4667 


2 .1429 


4881 2.0488 


5099 1. 9612 


5321 


1.8794 


5547 


1.8028 


59 


2 


4670 


2.1413 


4885 2.0473 


5103 1.9598 


5325 


1. 8781 


5551 


1. 8016 


58 


3 


4674 


2.1396 


4888 2.0458 


5106 1.9584 


5328 


1.8768 


5555 


1 . 8003 


57 


4 


4677 


2.1380 


4892 2 . 0443 


5110 1.9570 


5332 


1.8755 


5558 


I-799I 


56 


5 


4681 


2.1364 


4895 2.0428 


5114 1.9556 


5336 


1. 8741 


5562 


1.7979 


55 


6 


4684 


2.1348 


4899 2.0413 


5117 1-9542 


5340 


1.8728 


5566 


1 .7966 


54 


7 


4688 


2.1332 


4903 2.0398 


5121 1.9528 


5343 


1. 8715 


557o 


1-7954 


53 


8 


4691 


2.1315 


4906 2.0383 


5125 1. 9514 


5347 


1 .8702 


5574 


1.7942 


52 


9 


4695 


2 .1299 


4910 2.0368 


5128 1.9500 


5351 


1.8689 


5577 


1.7930 


5i 


10 


4699 


2.1283 


4913 2.0353 


5132 1.9486 


5354 


1.8676 


558i 


I-79I7 


50 


ii 


4702 


2.1267 


4917 2.0338 


5136 1.9472 


5358 


1.8663 


5585 


1 ■ 7905 


49 


12 


4706 


2.1251 


4921 2.0323 


5139 1.9458 


5302 


1.8650 


5589 


1-7893 


48 


13 


4709 


2.1235 


4924 2.0308 


5143 1.9444 


5366 


1.8637 


5593 


1. 7881 


47 


14 


47i3 


2.1219 


4928 2.0293 


5147 I.9430 


5309 


1.8624 


5596 


1.7868 


46 


15 


4716 


2.1203 


4931 2.0278 


5150 1. 9416 


5373 


1.8611 


5600 


1.7856 


45 


16 


4720 


2.1187 


4935 2.0263 


5154 1.9402 


5377 


1.8598 


5604 


1 . 7844 


44 


17 


4723 


2.1171 


4939 2.0248 


5158 1.9388 


538i 


1.8585 


5608 


1.7832 


43 


18 


4727 


2.1155 


4942 2.0233 


5161 1.9375 


5384' 


1.8572 


5612 


1 .7820 


42 


iQ 


473i 


2.1139 


4946 2.0219 


5165 1. 9361 


5388 


1-8559 


5616 


1 . 7808 


41 


20 


4734 


2. 1 1 23 


4950 2.0204 


5169 1.9347 


5392 


1.8546 


5619 


1.7796 


40 


21 


4738 


2 .1107 


4953 2.0189 


5172 1.9333 


5396 


1-8533 


5623 


1-7783 


39 


22 


4741 


2 .1092 


4957 2.0174 


5176 1. 9319 


5399 


1.8520 


5627 


1. 7771 


38 


23 


4745 


2 .1076 


4960 2.0160 


5180 1.9306 


5403 


1.8507 


5631 


1-7759 


37 


24 


4748 


2 . 1060 


4964 2.0145 


5184 1.9292 


5407 


1.8495 


5635 


1-7747 


36 


35 


4752 


2 . 1044 


4968 2.0130 


5187 1.9278 


54i 1 


1 . 8482 


5639 


1-7735 


35 


26 


4755 


2.1028 


4971 2.0115 


5191 1.9265 


5415 


1 . 8469 


5642 


1.7723 


34 


27 


4759 


2.1013 


4975 2.0101 


5195 1. 9251 


54i8 


1.8456 


5646 


1.7711 


33 


28 


4763 


2 .0997 


4979 2.0086 


5198 1.9237 


5422 


1 . 8443 


5650 


1.7699 


32 


29 


4766 


2.0981 


4982 2.0072 


5202 1.9223 


5426 


1 . 8430 


5654 


1.7687 


31 


30 


4770 


2.0965 


4986 2.0057 


5206 1. 9210 


5430 


1. 8418 


5658 


I.7675 


30 


31 


4773 


2.0950 


4989 2 . 004 2 


5209 1. 9196 


5433 


1 • 8405 


5662 


1.7663 


29 


32 


4777 


2.0934 


4993 2.0028 


5213 1. 9183 


5437 


1.8392 


5665 


1-7651 


28 


33 


478o 


2 .0918 


4997 2.0013 


5217 1. 9169 


5441 


1.8379 


5669 


1.7639 


27 


34 


4784 


2 . 0903 


5000 1 . 9999 


5220 1. 9155 


5445 


1.8367 


5673 


1.7627 


26 


35 


4788 


2.0887 


5004 1 . 9984 


5224 1. 9142 


5448 


1-8354 


5677 


1-7615 


25 


36 


4791 


2 .0872 


5008 1.9970 


5228 1. 9128 


5452 


1. 8341 


5681 


1 . 7603 


24 


37 


4795 


2.0856 


5011 1.9955 


5232 1.9115 


5456 


1.8329 


5685 


1 -7591 


23 


38 


4798 


2 . 0840 


5015 1. 9941 


5235 1.9101 


5460 


1. 8316 


5688 


1-7579 


22 


39 


4802 


2.0825 


5019 1.9926 


5239 1.9088 


5464 


1 • 8303 


5692 


I-7567 


21 


40 


4806 


2 . 0809 


5022 1. 9912 


5243 1.9074 


5467 


1 .8291 


5696 


1-7556 


20 


41 


4809 


2.0794 


5026 1.9897 


5246 1. 9061 


5471 


1.8278 


57oo 


1-7544 


19 


42 


4813 


2.0778 


5029 1.9883 


5250 1.9047 


5475 


1.8265 


5704 


1-7532 


18 


43 


4816 


2.0763 


5033 1.9868 


5254 1.9034 


5479 


1-8253 


57o8 


1.7520 


17 


44 


4820 


2.0748 


5037 1.9854 


5258 1.9020 


5482 


1.8240 


5712 


1.7508 


16 


45 


4823 


2.0732 


5040 1 . 9840 


5261 1.9007 


5486 


1.8228 


5715 


1 . 7496 


15 


46 


4827 


2.0717 


5044 1.9825 


5265 1.8993 


5490 


1. 8215 


5719 


1.7485 


14 


47 


4831 


2.0701 


5048 1. 981 1 


5269 1.8980 


5494 


1 .8202 


5723 


1 ■ 7473 


13 


48 


4834 


2.0686 


5051 1.9797 


5272 1.8967 


5498 


1 .8190 


5727 


1. 7461 


12 


49 


4838 


2 .0671 


5055 1.9782 


5276 1.8953 


5501 


1. 8177 


5731 


1 . 7449 


11 


50 


4841 


2.0655 


5059 1.9768 


5280 1.8940 


5505 


1. 8165 


5735 


1-7437 


10 


51 


4845 


2 . 0640 


5062 1.9754 


5284 1.8927 


5509 


1-8152 


5739 


1.7426 


9 


52 


4849 


2.0625 


5066 1.9740 


5287 1. 8913 


5513 


1 .8140 


5743 


I-74I4 


8 


53 


4852 


2 . 0609 


5070 1.9725 


5291 1.8900 


5517 


1 .8127 


5746 


1 . 7402 


7 


54 


4856 


2.0594 


5073 1.9711 


5295 1.8887 


5520 


1. 8ns 


5750 


1 -7391 


6 


55 


4859 


2.0579 


5077 1.9697 


5298 1.8873 


5524 


1. 8103 


5754 


1-7379 


5 


56 


4863 


2.0564 


5081 1.9683 


5302 1.8860 


5528 


1 .8090 


5758 


1.7367 


4 


57 


4867 


2.0549 


5084 1 . 9669 


5306 1.8847 


5532 


1.8078 


5762 


1-7355 


3 


58 


4870 


2.0533 


5088 1.9654 


5310 1.8834 


5535 


1 . 8065 


5766 


1-7344 


2 


59 


4874 


2.0518 


5092 1 . 9640 


5313 1.8820 


5539 


1.8053 


577o 


1-7332 


1 


60 


4877 


2.0503 


5095 1.9626 


5317 1.8807 


5543 


1 . 8040 


5774 


!-732I 





' 


cot 


tan 


cot tan 


cot tan 


cot 


tan 


cot 


tan 


t 


64° 


63° 


62° 


61° 


60° 



18] SOLUTION OF RIGHT TRIANGLES BY NAT. FUNCTIONS 41 

3. Find sin 28 30' 24", tan 6i° 24' 36", cos 6o° 30' 25". 

Ans. 0.4773, 1-8349, °-49 2 3- 

4. Find cos 25 50' 10", cot 28 25' 50", tan 27 00' 30". 

Ans. 0.9001, 1. 8471, 0.5097. 

5. Find sec 25°35', esc 25°28\ Ans. 1.1087, 2.3256. 
Solve the following right triangles: 

6. Given an acute angle A = 26 15' and the hypotenuse c— 35.0; 
to find the remaining parts. 

Ans. B = 63 45', a = 15.5, b = 31.4. 

7. Given A = 2 8° 50' and the side opposite, a = 150. 

Ans. B = 6i° io', b = 272, c = 311. 

8. Given ^4 = 63°4o / 3o // and the side adjacent, b = 363. Find 
a and c and check your results. 

18. To Find the Angle Less than 90° Corresponding to a 
Given Natural Function. 

a. When the function is one of the numbers given in the table, the 
angle is found by taking the number of degrees which stands at the 
head or foot of the column according as the name of the function 
appears at the head or the foot of the column in which the number 
is found, and the number of minutes at the left or right end of the 
line in which the number is found according as the degrees have been 
taken from the top or the bottom of the column. 

Example i. Find the angle whose sine is 0.4633. 

Solution. We find the number 4633 in the column which has 27 
written at its head and 62 at its foot, and in the line which begins 
with 36' and ends with 24' (see specimen page, p. 40). Since the 
given number is a sine, and the name " sin " appears at the head of 
the column in which 4633 is found, we take the degrees from the top 
of the column and the minutes from the left of the line in which 4633 
is found, that is, 

the angle whose sine is 0.4633, or sin -1 0.4633 = 27 36'. 

Example 2. Find the angle whose cosine is 0.4633. 
Solution. This time the name of the function appears at the 
bottom of the column in which 4633 is found, hence 

the angle whose cosine is 0.4633, or cos -1 0.4633 = 62 24'. 



42 PLANE TRIGONOMETRY [chap, in 

b< When the junction is not given in the table, we find the corre- 
sponding angle by reversing the process by means of which we find 
the function when the angle is given. 

Example 3. tan x = 0.5492, to find x. 

Solution. The number 5492 is not found in the table of natural 
tangents and cotangents, but two other numbers are found (see 
specimen page), namely, 5490 and 5494, one of which is a little 
smaller, the other a little larger than the given number. 

0.5490 = tan 28°46' 
0.5494 = tan 28°47 / . 

It is plain, therefore, that x, the angle whose tangent is 0.5492, is 
somewhere between 28 46' and 28°47 / . Applying the principle of 
proportional parts we have, 

x — 28°46' tan x — tan 28 46' 

28°47' - 28°46 / ~ tan 28°47' ~ tan 28°46' 
that is, 

x — 28°46 r _ 0.5492 — 0.5490 _ 0.0002 __ 2 _ 1 
i / or6o // 0.5494 — 0.5490 0.0004 4 2 

and solving for x, 

x = 28°46' + i of 60" - 28° 4 6 , 3o ,/ . 

It is not necessary to go through all this work each time. All we 
need to remember is that the smaller of the two angles between 
which x lies must be increased by 

- of 60", 
D 

where 

D is the difference (without regard to the decimal point) between 
the functions of the two angles between which x lies, 

d is the difference (without regard to the decimal point) between 
the function of the smaller angle and the given function. 

Example 4. tan x = 1.9887, to find x. 
Solution. From the table (specimen page) 

1.9883 = tan63°i8 / 
1.9897 = tan63° 19' 
D = 19897 - 19883 = 14, 
d = 19887 - 19883 = 4, 
x = 63 18' + T 4 4 of 60" = 63 18' 17". 



ig] SOLUTION OF RIGHT TRIANGLES BY NAT. FUNCTIONS 43 

Example 5. cosx = 0.4767, to find x. 
Solution. From the table 

0.4769 = cos6i°3i / 

0.4766 = cos6i°32 r 
D = 4769 - 4766 = 3, 
d = 4769 - 4767 = 2, 
x= 6i°3i , + f of 60" = 6i°3i / 4 o". 

19. Accuracy of Results. When an angle has been obtained from 
a four-place table (a table giving four places of decimals only), the 
number of seconds in the angle found cannot be relied upon with 
certainty. This is best shown by considering a special example, as 
Example 4 above. 

Since the tangents are given to four places only,- the actual value 
of tan 63 18' is not necessarily 1.9883, but may have any value 
between 1.98825 and 1.98835. 

Similarly the actual value of tan 63 ° 19' may be any number be- 
tween 1.98965 and 1.98975. 

Hence D, the difference between tan 63 19' and tan 63 18', is not 
necessarily 0.0014, but may be any number 

less than 1.98975 — 1.98825 =0.00150, 
and greater than 1.98965 — 1.98835 = 0.00130. 

Again, tan x, four places only being given, may have any value 
between 1.98865 and 1.98875, so that d, the difference between tan x 
and tan 63 18', may be any number 

less than 1.98875 — 1.98825 = 0.00050, 
and greater than 1.98865 — 1.98835 = 0.00030. 

Hence, 

— of 60" may be any number less than^ — of 60" = 23", 
D 130 

and greater than-^ — of 60" = 12". 
150 

The conclusion is that the result 63 18' 17" previously given is 
uncertain by as much as 6". 



44 PLANE TRIGONOMETRY [chap, in 

From the example just given, it appears that the amount by 
which the result obtained from a table is uncertain depends upon 
the difference D, which varies not only for different functions of the 
same angle, but also for the same function of different angles. No 
general rule can be laid down to cover the amount of uncertainty in 
all cases. If absolute certainty in the number of seconds is required, 
a seven-place table should be used, giving the values of the functions 
from second to second for small angles, and for intervals of 10" for 
larger angles. 

When a four-place table is used, and no special consideration is 
given to the nature of the differences involved, the number of seconds of 
an angle obtained by interpolation cannot be relied upon. 

The following rules will be of some aid to beginners: 

i. An angle less than 45 can be obtained more accurately from a sine 
than from a cosine, while an angle greater than 45 can be obtained 
more accurately from a cosine than from a sine. 

This is because the sines of angles less than 45 ° vary more than the 
cosines, and at the same time the principle of proportional parts is 
more nearly true for sines of small angles than for cosines, while the 
opposite is true for angles greater than 45 . 

2. Very small angles can be obtained with greater accuracy by inter- 
polation from tangents than from cotangents, while angles near go? can 
be obtained with greater accuracy from cotangents than from tangents. 

The reason for this lies in the fact that the principle of propor- 
tional parts ceases to apply to cotangents of small angles and to 
tangents of angles near 90 . 

Exercise 12 
Find the following angles: 

1. sin -1 0.4904, cos -1 0.4904, tan -1 1.8940, cot -1 1.8940. 

Ans. 2 9 °22', 6o° 38', 62 10', 27 50'. 

2. sin -1 0.4267, cos -1 0.4900, tan -1 2.1036, cot -1 0.5644. 

Ans. 25 15' 30", 6o° 39' 30-, 64° 34 r 30", 6o° 33' 30". 

3. cot -1 2.1441, tan -1 0.4737, tan -1 1.7611, cot -1 1.7611. 

Ans. 25°oo / 15", 25°2o , 45", 6o°2 4 , 4o", 2 9 °35 , 20 ,/ . 

4. Show that sin" 1 0.4250 + sin -1 0.9052 = 90 . 

5. Show that sin -1 0.4488 + sin -1 0.4746 = 55 . 



so] SOLUTION OF RIGHT TRIANGLES BY NAT. FUNCTIONS 45 

6. If tan -1 0.5000 + tan -1 x = 87°34 / , find x. 

Ans. x =1.8040. 

7. In a right triangle one side b = 4.63 and the hypotenuse c = 10.0. 
Find the included angle A. Ans. A = 62 25'. 

(Suggestion, cos A = b/c.) 

8. In a right triangle the hypotenuse c = 35.00 and the side 
a = 31.29. Find the angle opposite a, and the third side b. 

Ans. A = 63 23', b = 15.68. 

9. Two sides of a right triangle are a = 475.0, b = 237.5. Find 
the angles A and B and the hypotenuse c. 

Ans. A = 63 26', B = 26 34', c = 531.1. 

10. Determine the uncertainty in the number of seconds of 
tan -1 2. 1 21 1 as given by a four-place table, assuming .the principle 
of proportional parts. Ans. tan -1 2.1211 = 64°45 / 3o // ± 2" . 

20. Solution of Right Triangles by Natural Functions. In order 

to solve a right triangle, two parts must 
be given besides the right angle, and one 
of these parts must be a side. 

Let ABC be any right triangle, A, B, C 
the angles, and a, b, c the corresponding 
Fi S- 2 3- sides. Then 

- = sin;!, (1), - = cos A, (2), - = tan ,4, (3). 
c c b 

These three equations are sufficient for the solution of any right 
triangle, for, — 

If one of the given parts is an angle, we may call this angle A, and 
the other given part must be either a, b, ore; 

I. Given A and a\ then (3) gives b, and (1) gives c. 
then (2) gives c, and (3) gives a. 
then (1) gives a, and (2) gives b. 

If the given parts are both sides, there are three more cases, — 
IV. Given a and b; then (3) gives A, and c is found as in I or II. 
V. Given b and c; then (2) gives A, and a is found as in II or III. 
VI. Given c and a; then (1) gives A, and B is found as in III or I. 
Since A and B are complementary, B may always be found from 
the relation A -\- B — 90 . 




II. Given A and b 
III. Given A and c 



4 6 



PLANE TRIGONOMETRY 



[chap, ni 



It is not necessary to consider the various cases of right triangles 
separately, for the following simple rule governs all cases: 

Employ that trigonometric function of the angle which involves the 
two sides under consideration. The two sides may both be given, or 
one may be given and the other required to be found. 

Example i. Given one side of a right triangle equal to 418.5, 
and the hypotenuse equal to 614.0; required the other parts. 

Solution. Denote the given side by a or b, let us say by b, and the 
hypotenuse by c. We then have 

Required A =47° 02', 
B =42° 58', 
a = 449-3- 



Given b = 418.5, 
c = 614.0. 

1. To find A. 
We have 

and from the table 




cos A = h - = 41^ = 0.6816, 



2. To find B. 

3. To find a. 
From (3), 
From the table 

Multiplying, 



Fig. 24. 
A = cos -1 0.6816 = 47 02 r . 

A-\- B = 90 , hence B = 90° 



614.0 



47 02' 



42° 58'. 



a = b tan A. 
tan A = tan 47°o2 r = 1.0736 
b = 418.5 V 



a = 449.3 

4. Check. To guard against possible mistakes, the answers should 
be tested by some formula which has not already been used in ob- 
taining the answers. Now in solving a right triangle we never use 
more than two of the three formulas (1), (2), (3), hence the third 
may always be used as a check. In the present problem, (2) was 
used in finding A, and (3) in finding a, hence we may test our re- 
sults by using (1), that is, if our results are correct they should 
satisfy the relation (1), or 

a = c sin A 
sin A = sin47°o2' = 0.7318 
c = 614 

Multiplying, we get a = 449-3> 

which agrees with the value of a as determined above. 

Note. We might have used the relation a 2 + b 2 = c 2 as a check, but this would 
have required more work. 



2o] SOLUTION OF RIGHT TRIANGLES BY NAT. FUNCTIONS 47 



Example 2. Given one angle of a right triangle equal to 36 50', 
and the hypotenuse equal to 3.12; to solve the triangle. 

Solution. Denote the angle by A, and the hypotenuse by c, then 
we have, — 

Given A = 36 50', 
c= 3.12. 



1. To find a. 

From (1), 
From the table, 

Multiplying, 

2. To find b. 

From (2), 
From the tables, 




Required a = 1.87, 
b = 2.50, 
B = 53° 10'. 



Fig. 25 



a = c sin A. 
sin A = sin 36 50'= 0.5995 
c = 3.12 

a = 1.870 



b = c cos A . 
cos A = cos36°5o / = 0.8004 

c = 3-* 2 

b = 2.497 



Multiplying, 

3. To find B. 

A + B = 90 , hence B = go° - 36 50' = 53 10'. 

4. Check. Having already used (1) and (2), we use (3). 

a = b tan A . 
From the tables, tan A = tan36°5o /= 0.7490 

b = 2.497 

Multiplying, we get a = 1.870, 

which agrees with the value of a as obtained above. 

Instead of referring, to equations (1), (2), and (3), it is better to 
write down from the triangle under consideration that ratio which 
is needed to solve for a particular side or angle. The method will be 
sufficiently clear from an example. 1 

Example 3. Given = 15 25', k Required 6 = 74 35', 

m = 345. V k n = 1251, 

Solution. / p = I2 g8. 

~i. Tofindfl. e + cf> = go°, L 

= 9 o° -15° 25' =74° 35'. Fig. 2 6. 
2. To find n. Since </> and m are given, we must use that function 
of <f> which involves n and the given side m. 



m 
n 



= tsmcj), or n = m cot cf> = 345 X 3.6264 = 1251. 1. 



48 PLANE TRIGONOMETRY [chap, in 

3. To find p. We use that function of <j) which involves p and 
the given side m. 

— = sin 9, or ^ = - — - = °^ J = 1298. 
^ sin 9 0.2658 

4. Check. If our results for 8, n, and p are correct, we should have 

ft 

- = sin 0, or n = p sin = 1298 X 0.9640 = 125 1.3. 

P 

In this case there is a slight discrepancy between the result of the 
check and the value of n as found in 2. This is due to the fact that 
we have given the value of n to five places while the table gives but 
four places. The values to four places agree exactly. 

Exercise 13 

Solve the following right triangles. Check your results when the 
answers are not given. 

1. Given a = 10.00, A = 25 ; find b = 21.45, c = 23.66. 

2. Given b = 256, A =36° 30'; find a, B, and c. 

3. Given c = 350,^ = 56°45'; find B = 33° 15', a= 293, b =192. 

4. Given a = 346, B = 50 ; find the other parts. 

5. Given c = 45.7, B = 44 50'; find a = 32.4, 6 = 32.2. 

6. Given & = 13.5, B = 28°4o'; solve the triangle. 

7. Given a = 170, b = 350; find A = 25 54', B = 64 06', c = 389. 

8. Given a = 0.81, c = 2.54; solve the triangle. 

9. Given b = 6.57, c = 10.6; find ^4 = $i° 42 f ,B = 38 18', a = 8.32. 

10. Find the altitude of an isosceles triangle whose base is 368, 
and whose equal sides make an angle of 64 . Ans. 294. 

11. Find the perimeter and area of a regular pentagon inscribed 
in a circle whose radius is 10. Ans. 58.78, 237.76. # 

12. Show that the area of any right triangle is equal to 
either one of the expressions J be sin A or \ ac sin B. 

13. In the right triangle ABC, Fig. 27, AB = 338, 
angle B = 40 ; show how to find the length of the 
median AM, the length AS of the bisector of the 
angle A, and the angle included between these two. 




2i] SOLUTION OF RIGHT TRIANGLES BY NAT. FUNCTIONS 49 

14. An oblique triangle ABC has AB =120, BC= 150, and the 
angle B = 67°3o'; solve the triangle. 

Ans. AC = 152, A = 65°4i', C = 46°4 9 '. 
(Suggestion. Drop a perpendicular from A or C to the opposite 
side, dividing the triangle into two right triangles.) 

15. Given one side of an oblique triangle equal to 57.3, and the 
adjacent angles equal to 35 45' and 75 30' respectively; find the 
remaining parts. Ans. 59.5, 35.9, 68°45'. 

(Suggestion. Draw the altitude from a vertex adjacent to the 
given side.) 

21. Right Triangles Having a Small Angle. 
Given the hypotenuse c and a side b of a right triangle, to solve the 
triangle in case b and c are nearly equal. 
The angles A and B are given by the relation / 

cos A = - = sinB. 

Fig. 28. c 

It is apparent from the figure that if b is nearly equal to c, angle A 
must be very small and angle B must be nearly equal to 90 . . 

By examining the table of natural sines and cosines it will be seen . 
that for small angles the cosines are so nearly equal that there is no >J 
difference at all in the first four places, and similarly the sines of 
angles near 90 are nearly equal. Thus, so far as the table shows, 
all angles from o° to o° 34' have the same cosine, and likewise all 
angles between o°34' and o° 60', between i°oo' and i° 16', etc. It 
follows that a small angle cannot be accurately found from its cosine ', 
nor an angle near QO° from its sine. 

To avoid using the cosine, the following formula is used whenever J 
the given parts b and c are nearly equal, — 




2 V c -\- b 

Proof. Let ABC be any right triangle, b the base, a the altitude, 
c the hypotenuse. Produce CA to 0, making 
AO = c. Join O and B. Now angle .4 (= CAB) 
= angle AOB + angle ABO, hence since tri- 
angle AOB is isosceles, angle A = twice the 

angle AOB, that is, angle AOB = — , and Flg ' 2g ' 

2 




2 c^-b c4-b V^-4-^ 2 V, 



c-b 
2 c+b c+b V (c H- £) 2 V c + b 



50 PLANE TRIGONOMETRY [chap, m 

Example i. Given b = 25.7, c = 26.8; to find A, B, and a. 
Solution. c — b = 26.8 — 25.7 = 1.1, 

c + b = 26.8 + 25.7 = 52.5. 

tan —~=y - 1 — = V 0.020952 = 0.1447, 

- = 8° 14', A = 16 28', 
2 

B = go°- 16 28' = 73 32'. 

a = b tan ^4 = 25.7 X 0.2956 = 7.6. 

Check. a 2 = c 2 -b 2 = (c- b) (c + 6). 

(c- ft)(c+&) = 1.1 X 52.5 =57.75. 

Exercise 14 

1. Given b = 4.75, c = 5.25; solve the triangle. 

Ans. A = 25 13', B = 64 47', a = 2.24. 

2. Given a = 9.6, c = 10.4; solve the triangle and check your 
results. 

3 . Show that tan — = 



2 a 



A _Jc + b 



4. From Fig. 29 show that sin — = \l •, cos— = \/ : 

2 V 2C 2 V 2 C 

22. Historical Note. The use of tables of natural functions 
dates back to antiquity. In the second century B.C., Hipparchus, a 
Greek astronomer and mathematician, constructed tables of chords 
(double sines) which answered the same purpose as a table of sines. 
The tables of Hipparchus have been lost. The oldest table now 
extant is that of Ptolemaus (second century A.D.), giving double 
sines from minute to minute with an accuracy which in our system 
of numeration would be expressed by five places of decimals. 

Hindu mathematicians as early as the fifth century A.D. were in 
possession of a small table which they memorized, very much as we 
memorize our multiplication table. Values not given in the table 
were computed from memory as occasion required, by means of a 
formula put in verse. 

The first table which approached in extent and arrangement the 
tables now in use is the " Canon doctrinae triangulorum " of Rheti- 



23] SOLUTION OF RIGHT TRIANGLES BY NAT. FUNCTIONS 51 

cus (1551). This table gives each of the six functions for intervals 
of 10" from o to 45. Like present tables, the degrees and seconds 
proceed from top to bottom in the left marginal columns, while the 
complementary angles proceed from bottom to top in the right 
marginal columns. Later, Rheticus prepared a second table for 
which the sines of angles for intervals of 10" were computed to 15 
places of decimals, though only 10 places were retained in the " Opus 
Palatum m," the name under which the table was published. 

Rheticus' table contained numerous errors, which were largely re- 
moved by Pitiscus, an indefatigable arithmetician of the seventeenth 
century. In addition to revising the existing tables he computed 
anew the sines of angles from o° to 7 for intervals of i' to from 20 
to 25 places of decimals. Pitiscus 's improved tables were published 
in 1 6 13 under the title " Thesaurus mathematicus." These tables 
formed the basis of all subsequent tables, until the discovery of 
improved methods of computation in recent times has made it com- 
paratively easy to check old tables or to compute new ones. 

23. Review. 

1. (a) Define trigonometry, explain the etymology of the word 
and tell how the science originated, (b) Define in words the sine, 
cosine, and tangent of an acute angle, (c) Define the secant, co- 
secant, and cotangent of an angle, (d) What is meant by the versine 
and coversine of an angle ? 

2. (a) Name three pairs of functions such that in each pair either 
is the cofunction of the other, (b) Explain the origin of the terms 
cosine, cosecant, and cotangent, (c) Prove that sin A = cos (90 — A), 
cos A = sin (90 — A), tan A = cot (90 — A), (d) Express sin 76 40' 
as a function of an angle less than 45 °. 

3. (a) Construct the following angles: sin -1 J, cos -1 0.4, tan -1 0.5, 
cot -1 3. (b) Give from memory the values of the sine, cosine, and 
tangent of each of the following angles: o°, 30 , 45 , 6o°, 90 . 
(c) Draw a figure and deduce the functions of 30 and 45 . 

4. (a) Name three pairs of functions such that in each pair either 

function is the reciprocal of the other, (b) Prove the relations 

sin 2 A + cos 2 A = 1, tan 2 A + 1 = sec 2 A, cot 2 A + 1 = esc 2 A, 

sin A 3 

tan A = . (c) Given sin A = - , find each of the other func- 

cos A 5 

tions of A . (d) Express each of the functions of A in terms of tan A . 



52 PLANE TRIGONOMETRY [chap, in 

5. (a) Reduce to its simplest form the expression 

cos X , sin X 



1 — tan X 1 — cot X 
(b) Prove the identities, — 

sec A -f esc A _ tan A + 1 __ 1 + cot ^4 
sec ^4 — csc ^4 tan A — 1 1 — cot ^4 

6. (a) What is meant by a table of natural functions? (b) Which 
functions increase and which decrease as the angle increases from 
o° to 90 ? (c) What is meant by the term " interpolation" ? by 
the " principle of proportional parts" ? (d) How would you find the 
secant of a given angle ? (e) Can a small angle be found more 
accurately from its sine or from its cosine ? Why ? (/) Can a small 
angle be found more accurately from its tangent or from its cotan- 
gent ? Why ? 

7. (a) There are four different cases of right-triangle problems 
according as the given parts are: I. The hypotenuse and an angle. 
II. One side and an angle. III. One side and the hypotenuse. 
IV. Two sides. Show how to solve each case. 

A lc — b 

8. Prove the formula tan — = V / , and write down a similar 

2 V c+ b 

formula for tan — . 

2 

9. A flagstaff AB stands on top of a tower. To determine its 
height, a distance OP 500 feet long was measured off in a horizontal 
direction from the foot of the tower. At P the angles OP A and 
OPB were measured, and were found to be 22 and 28 respectively. 
Find the length of the flagstaff by natural functions, also by the 
graphic method, and compare your results. 



CHAPTER IV 
LOGARITHMS 

24. Definition of Logarithm. The numbers representing the 
given parts of a triangle and other numbers obtained by careful 
measurement usually contain three or four significant figures and 
sometimes five, six, or it may be even seven significant figures. 
Multiplication, division, the extraction of roots and raising to powers 
of such numbers by the ordinary methods, require a great deal of 
tedious labor, most of which may be avoided by using another 
method of computation, known as the method of logarithms. This 
method requires the use of tables, by the aid of which multiplica- 
tion of two or more numbers is accomplished by adding certain 
other numbers found in the tables. Similarly division is reduced to a 
mere subtraction of two numbers found in the tables. To raise to a 
power or to extract a root of a number, the corresponding number in 
the table is multiplied or divided by the index Of the power or root.* 

The method of logarithms presupposes that when some positive 
number, different from unity, has been chosen, every other positive 
number can be expressed as some power (integral, fractional, nega- 
tive, or incommensurable) of this number. Thus, if a is some posi- 
tive number not equal to one, and N any positive number, we assume 
that a number x can always be found such that 

a x = N. 

This number x is called the logarithm of N to the base a, and is 
usually written 

X = l0g a N, 

in words: 

* The usefulness of the method of logarithms may be anticipated from the 
testimony of Laplace, the great French astronomer, who said the method of log- 
arithms " by reducing to a few days the labors of many months, doubles, as it were, 
the life of an astronomer, besides freeing him from the errors and disgust insepa- 
rable from long calculation." The advantages which the use of logarithms offers 
to the astronomer are shared, of course, by all others who deal much with numeri- 
cal calculation. 

53 



54 PLANE TRIGONOMETRY [chap, iv 

The logarithm * of a number to a given base is the exponent of the 
power to which the base must be raised to produce the number. 

For example, since 

io 2 = ioo, 2 = logio ioo; 

io 3 = iooo, 3 = logio iooo; 

io* = 3.1623, § = Iogio3.i623; 

io -1 = 0.1, — 1 = logio 0.1 ; 
(0.5)2 = 0.25, 2 = logo.50.25; 

2 7 § = 9, ! = log 2 7 9; 

a s = 4 3 r= 8 2 = 64, 6 = log 2 64, 3 = log 4 64, 2 = logs 64. 

25. Fundamental Laws Governing Logarithms. Since loga- 
rithms are exponents, the laws of logarithms are the same as the 
laws of exponents. Now the laws of exponents are, — 

(a) a* • a ?/ = *»+ v , 

x is the logarithm (exponent) of the first factor, 
y is the logarithm (exponent) of the second factor, 
x + y is the logarithm (exponent) of the product; 
hence, 

The logarithm of the product of two factors is equal to the sum of the 
logarithms of the factors, or 

log P = log M + log N. 

-,logP = logL + logif + logiy+'v . 
log 30= log 2 + log 3 + log 5. 

x is the logarithm (exponent) of the dividend, 
y is the logarithm (exponent) of the divisor, 
x — y is the logarithm (exponent) of the quotient ; 
hence, 

* From logos = ratio, and arithmos = number, so called by the Scotch mathe- 
matician John Napier, one of the inventors of logarithms, because, as originally 
conceived of, logarithms were a set of numbers, as 

h, k, I, m, n, etc., 
corresponding to a second set a h , a k , a 1 , a m , a n , etc., 

the second set being so chosen that the ratio between any consecutive two is the 
same. 



If 


P 


= M 


-N, 


Similarly, 








If 


P 


= L- 


M -N • 


Thus log 1 


5 = 


log 3 


+ log5, 


(b) a* ^ 


- a ?/ 


= a* 


-y 



26] . LOGARITHMS 55 

The logarithm of the quotient of two numbers is equal to the logarithm 
of the dividend diminished by the logarithm of the divisor, or, 

If Q = M + N, logQ = logM-logN. 

Thus log| = log5-log3. 

(c) (a")" = a wac , 

x is the logarithm of the quantity which is to be raised to the 
nth power, nx is the logarithm of the resulting power; hence, 

The logarithm of any power of a number is equal to the logarithm of 
the number multiplied by the index of the power to which it is to be raised, 
or 

If P = N n , logP = nlogN. 

Thus log3 5 =5log3. 

(d) yp = a-, 

x is the logarithm of the quantity whose nth root is to be extracted, 

x 

- is the logarithm of the resulting root ; hence, 

n 

The logarithm of any root of a number is equal to the logarithm of 
the number divided by the index of the root to be extracted, or 

If P=W, lo g P = l °&X. 

n 

Thus log V17 = ^log 17. 

26. Logarithms of Special Values. 

(a) a 1 = a, hence log a = 1, that is, 
The logarithm of the base is 1. 

(b) Any number divided by itself is 1, but by the law of expo- 
nents a -s- a = a°, so that a° = 1, and therefore log a 1 = 0, that is, 

The logarithm of 1 to any base is o. 

(c) By Art. 25 (b), log^- = log 1 — log N, and by (b) of this 
article, log 1 = 0, hence 

l ° g N = ~ logN > 
that is, 

The logarithm of the reciprocal of any number is equal to minus the 
logarithm of the number. 



56 PLANE TRIGONOMETRY [chap, iv 

Definition. The logarithm of the reciprocal of a number is called 
the cologarithm * of the number. 
Thus 

logio 10= i, hence logio T V = — i = cologio 10; 

logio ioo = 2, hence logio t<jo = — 2 = cologio 100; 

logio 3.1623 = i, hence logio — — = — i = cologio 3.1623. 

3.1623 

Since — - = M • — ; , we have 

N N 

log ^ = log M + log j- = log M + colog N, 

that is, 

The logarithm of the quotient of two numbers is equal to the logarithm 
of the dividend plus the cologarithm of the divisor. 

Exercise 15 

1. Given 5 3 = 125, 5 2 = 25, 5 1 = 5; write down the logarithms to 
the base 5, of 125, 25, 5, J, ^, T £y. 

2. Given log 4 2 = 0.5, logs 81 = 4, logio 3- l62 3 = °-5> l°g& a = c \ 
write down equivalent expressions free from the symbol log. 

Ans. 4 ' 5 = 2, 3 4 = 81, etc. 

3. Find logs 27, logio 10,000, log a a, logs wt, log a a*. 

i4/w. 3, 4, 1, -3, f. 

4. Express in terms of log 2, log 3, and log 5, the following: 

log 15, log — , log|, log 100, logW^, log _ I5 - 
36 V 2 ^/6o 

4*w. log 3 + log 5, log 2 + log 5 - log 3, log 5 - log 2 - log 3, 
2 (log 2 + log 5), I (log 3 + log s -log 2), T V log 2 + ^ log 3 + T % log 5. 

5. Given logio 2 = 0.30103, logio 3 = o.477 I 2 > lo gio 5 = 0.69897; 

find logio 4, logio 0.3, logio 0.75, logio ^5* logioV^-^. 

Ans. 0.60206, — 1 + 0.47712,! — 1 + 0.87506, 0.23299, 0.12224. 
* Whenever x + y = constant, x and y are said to be complements (more 

specifically arithmetic complements) of each other. Now log x + log — = log 1 

x 

= constant, hence log - is the complement of log x, which on contraction becomes 

colog X. 

t When 10 is the base, the logarithm is always written so that the fractional 
part is positive. 



27] LOGARITHMS 57 



6. If b = 184, c = 59; find the logarithm of 1/ . , and also of 

V b — c 

b 2 — c 2 . Ans, 0.14434, 4.48251. 

7. Show that the fractional part of a logarithm, to the base 10, is 
not changed if the number is multiplied by 10, or by a power of 10. 

8. Compute to four places of decimals the numbers whose loga- 
rithms to the base 10 are J, J, §, 3-. 

Ans. 3.1623, 1.7783, 2.1544, 1.4678. 

27. The Common System of Logarithms. Each different base 
determines a different system of logarithms. The system in com- 
mon use has 10 for its base, and is called the common * system of 
logarithms. Common logarithms have been carefully computed and 
tabulated, so that the logarithm of any number can be readily found 
by referring to the table, and, vice versa, if the logarithm of a 
number is known, the number itself can be found from the table. 

The advantage of the common system over other systems of 
logarithms consists in this: the fractional part of any common loga- 
rithm remains unchanged when the number is multiplied or divided 
by 10 or a power of 10. To see this, let us consider a special case, 
say the number whose logarithm is 0.5. 

io 05 = 10? = V 10 = 3.16228, therefore log 13.16228 = 0.5. 

Multiplying by 10, 

10 X io 05 = 10 X 3.16228, 

or io 1-5 = 31.6228, therefore log 31.6228 = 1.5. 

Similarly 

io 2.5 _ 216.228, therefore log 316.228 = 2.5; 

io 3.5 _ 2162.28, therefore log 3162.28 = 3.5; 

io 4.5 _ 21622.8, therefore log 31622.8 = 4.5; etc. 

Again, dividing by 10, we have, 

io - 5 _, * e 3.16228 

= 10 1 X io°- 5 = Q 

10 10 

or io -1+0 - 5 = 0.316228, log 0.316228 = — 1+ 0.5 = 9.5 — 10. 

* This system is also known as the Briggsian system, in honor of Henry Briggs 
of Oxford (1556-1630), who was the first one to compute and publish a table of 
logarithms to the base 10. 

f In the remainder of this chapter, and always when computations are con- 
cerned, if no base is expressed, the base 10 is understood. 



58 PLANE TRIGONOMETRY [chap, iv 

Similarly 

IO -2+o.5_ 0.0316228, log 0.0316228 = — 2 + 0.5 = 8.5—10; 
IO -3+o.5_ 0.00316228, log 0.00316228 = —3 + 0.5 = 7.5 — 10; 

and so on, the form 9.5 — 10 being introduced to avoid the plus 
sign between — 1 and 0.5. 

Now the logarithms of each of the numbers 3.16228, 31.6228, 
316.228, 3162.28, 31622.8, etc., 0.316228, 0.0316228, 0.00316228, etc., 
have the same fractional part, namely, 0.5, while the integral parts 
1, 2, 3, 4, etc., —1, —2, —3, etc., plainly depend upon the position 
of the decimal point. 

For convenience the integral part of the logarithm is called the 
characteristic of the logarithm, and the fractional part is called the 
mantissa. 

Thus, the logarithm of 316.228 is composed of the characteristic 2 
and the mantissa 0.5. 

The logarithm of 0.00316228 is composed of the characteristic —3 
and the mantissa 0.5. 

We have seen that the mantissa is independent of the position of 
the decimal point, that is, it is the same for all numbers composed 
of the same figures taken in the same order. 

28. Rule for the Characteristic. We shall now learn how the 
characteristic of a logarithm of a number may be determined before 
the logarithm is known. 

io° = 1, or log 1 = 0; 

io 1 = 10, or log 10 = 1; 

io 2 = 100, or log 100 = 2; 

io 3 = 1000, or log 1000 = 3; etc. 

Since log 1 = and log 10 = 1, 

every number between 1 and 10 has a logarithm between o and 1,* 
or, every number whose integral part has one digit has a logarithm 
whose characteristic is zero. 

Since log 10 = 1 and log 100 = 2, 

every number between 10 and 100 has a logarithm between 1 and 2, 

* This statement assumes that to the greater of two numbers corresponds the 
greater logarithm, that is, if M > N, log M > log N, a theorem which can be 
easily proven by elementary algebra^ 



28] LOGARITHMS 



59 



or, every number whose integral part has two digits has a logarithm 
whose characteristic is one. 

Similarly, every number between ioo and iooo has a logarithm 
between 2 and 3, or, every number whose integral part has three digits 
has a logarithm whose characteristic is two. 

Every number whose integral part has four digits has a logarithm 
whose characteristic is three, and so on. 

This gives us the following rule: 

I. The characteristic of the logarithm of any number greater than 1 
is one less than the number of digits in the integral part of the number. 

Thus, the integral part of the number 31622.8 consists of 5 digits, 
hence the characteristic of its logarithm is 4. 

The integral part of 3.16228 consists of 1 digit, hence the charac- 
teristic of its logarithm is o. 



Again, 



10 


— 


I, 




lO" 1 


= 


I 

IO 


O.I, 


IO" 2 


= 


I 

io 2_ 


O.OI, 


IO~ 3 


= 


I 

IO 3 


0.001, 


IO- 4 


= 


I _ 
io 4 ~ 


0.0001, 



or log 1 = 0; 

or log 0.1 = — 1; 

or logo.oi = — 2; 

or logo.ooi = — 3; 

or logo.oooi = —4; etc. 



Hence, every number between 1 and 0.1 has a logarithm between o 
and — 1, or, every fraction greater than 0.1 has a logarithm whose 
characteristic is — 1 . 

Similarly, every number between 0.1 and 0.01 has a logarithm be- 
tween — 1 and —2, or, every fraction greater than 0.01 but less than 
0.1 has a logarithm whose characteristic is —2. 

In like manner, every fraction greater than 0.001 but less than 0.01 
has a logarithm whose characteristic is —3, and so on. 

This gives us a second rule: 

II. The characteristic of the logarithm of any number less than 1 is 
a negative number one more than the number of ciphers between the 
decimal point and the first significant figure of the fraction. 



60 PLANE TRIGONOMETRY [chap, iv 

Thus, the fraction 0.00316228 has two ciphers between the decimal 
point and the first significant figure, hence the characteristic of its 
logarithm is — 3 or 7 — 10. 

The fraction 0.316228 has no cipher between the decimal point 
and the first significant figure, hence the characteristic of its loga- 
rithm is — 1 or 9 — 10. 

Exercise 16 

1. Write down the characteristics of the common logarithms of 
3 6 7> 3 6 -7> 3°7°, 3-°7> 0.000367, 0.367. 

Ans. 2, 1, 3, o, —4 or 6 — 10, — 1 or 9 — 10. 

2. log 635 = 2.80277; write down the logarithms of 6.35, 63500, 
0.635, 0.0000635. 

Ans. 0.80277, 4.80277, 9.80277 — 10, 5.80277 — 10. 

3. How many digits are there in the integral part of the number 
whose logarithm is 3.1567; 1.6533; 0.6831? 

4. To the number 57 corresponds the mantissa 75587; what is the 
number whose logarithm is 1.75587; 2.75587; 0.75587; 9.75587 — 10? 

Ans. 57; 570; 5.750.57. 

5. To the number 673 corresponds the mantissa 82802; find the 

logarithm of 673; of (673)2; °f ^°735 of V(673) 2 . 

Ans. 2.82802; 5.65604; 1.41401; 1.88535. 

6. log 3 = 0.47712; how many digits are there in 3 25 ; in 3 100 ; in 30 15 ; 
in 27 s ? Ans. 12; 48; 23; 8. 

7. Write down the cologarithms of the numbers in problem 2. 

Ans. 9.19723 - 10; 5.19723 - 10; 0.19723; 4-i97 2 3- 

8. log 5 = 0.69897; find the logarithm of i; of ^V? of 0.05; of 
V5; ofVi; of Vo^; oftyj. 

Ans. 9.30103 — 10; 8.60206 — 10; 8.69897 — 10; 0.34948; 
9.65052 — 10; 9.89966 — 10; 9.76701 — 10. 

9. To the number 3 corresponds the mantissa 47712, and to the 
number 7 corresponds the mantissa 84510; find the logarithm of 21; 

of f ; of V0.3 X 495 of y/i. 

Ans. 1.32222; 9.63202 — 10; 0.58366; 9.96362 — 10. 




ag] LOGARITHMS 6 1 

10. The formula for the amount (A) of a principal (P) put out on 
compound interest at (R) per cent for (t) years, the interest being 
compounded annually, is 



whence 



ioo/ 



io g ^ = io g p + nogfi + XY 



Find the number of digits in the amount of $i at compound 
interest at 6 per cent for ioo years, the mantissa corresponding to 
106 being .02531. Ans. 3 digits. 

29. Tables of Common Logarithms. Our next step is to learn 
how to use a table of logarithms in actual computation. 

Common logarithms, except those belonging to numbers which 
are integral powers of 10, cannot be exactly expressed in decimals. 
We must therefore omit all the figures after a certain decimal place. 
Just where to stop depends upon the accuracy desired. If some of 
the numbers which enter a given problem are the results of measure- 
ment, as is frequently the case, their accuracy will not ordinarily 
exceed four or five figures, consequently it will be useless to retain 
more than five figures in the decimal part of their logarithms. In 
other words, a table of logarithms which contains the mantissas to 
five places will answer for the solution of most practical problems 
in which approximate answers are all that is necessary or possible. 
Such a table is known as a five-place table of logarithms. 

The explanations which follow, and all the answers in this book 
obtained from logarithmic computation, are based upon a five-place 
table. 

When a five-place table is used, it is not worth while to retain 
more than five significant figures in the answers to the problems, 
and even then the last figure is not always exact. 

When more than five-place accuracy is required, and even when 
the fifth place must be known with certainty, larger tables must be 
used. Tables containing six places, seven places, eight places, ten 
places, eleven places, twenty places, and partial tables containing up 
to two hundred and sixty places, have been published. There are 
also smaller tables containing three and four places only. 

Tables containing more than seven places are seldom used, for seven- 



62 PLANE TRIGONOMETRY [chap, iv 

place tables meet practically every demand of present-day science. 
The results obtained by means of a seven-place table are as exact as 
the most careful measurements obtained by the most skillful observers 
by means of the most precise instruments under the most favorable 
conditions. 

30. To Find the Logarithm of a Given Number. 

(a) When the given number has four figures. The characteristic is 
found by the rule in Art. 28. To find the mantissa, enter that line 
of the table which begins with the number made up of the first three 
figures of the given number and take out that number of the line 
which is found in the column headed by the fourth figure of the 
given number. 

The number thus found constitutes the third, fourth and fifth 
figures of the required mantissa. The first and second figures are 
found in the column headed by o either in or above the line in which 
the third, fourth and fifth figures are found, except when these figures 
as given in the table are preceded by a star (*) in which case the first 
two figures are found in the next following line. 

Example i. From the specimen page of logarithms, page 63, we 
find 

mantissa log 6315 = 80037, hence log 6315 = 3.80037; . 
mantissa log 65.24 = 81451, hence log 65.24 = 1.81451; 
mantissa log 6.608 = 82007, hence log 6.608 = 0.82007. 

(b) When the given number has less than four figures. We know 
that the mantissa of the logarithm of a number is not changed if the 
number is multiplied or divided by 10 or by some power of 10. After 
we have determined the characteristic of a logarithm we may then 
annex as many ciphers to the given number as we need to make up 
four places and find the mantissa of this new number by case (a). 

Example 2. Find the logarithm of 64. 

The characteristic of log 64 is 1 . The mantissa of log 64 is the same 
as the mantissa of log 6400, which by case (a) is found to be 80618. 
Hence, 

log 64 = 1. 80618. 

(c) When the number has more than four figures. 

Example 3. Find the logarithm of 6425.4. 
The characteristic of log 6425.4 = 3. 



3°] 



COMMON LOGARITHMS OF NUMBERS 



63 



fi 











(SPECIMEN 


PAGE) 










N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D 


630 

631 
632 
633 


934 


941 


948 


955 


962 


969 


975 


982 


989 


996 


7 


80 003 
072 
140 


010 

079 
147 


017 
085 
154 


024 
092 
161 


030 
099 
168 


037 
106 
175 


044 
113 
182 


051 
120 
188 


058 
127 
195 


065 
134 
202 


7 
7 
7 


634 
635 
636 


209 
277 
346; 


216 
284 
353 


223 
291 
359 


229 
298 
366 


236 
305 
373 


243 
312 
380 


250 
318 
387 


257 
325 
393 


264 
332 
400 


271 

339 
407 


7 

7 
7 


637 
638 
639 

640 

641 
642 
643 


414 

482 
550 


421 

489 
557 


428 
496 
564 


434 

502 
570 


441 
509 
577 


448 
516 
584 


455 
523 
591 


462 

530 
598 


468 
536 
604 


475 
543 
611 


7 
7 

7 


618 


625 


632 


638 


645 


652 


659 


665 


672 


679 


7 


686 
754 
821 


693 

760 
828 


699 
767 
835 


706 

774 
841 


713 

781 
848 


720 


726 

794 
862 


733 

801 
868 


740 

808 
875 


747 
814 
882 


7 
7 
7 


644 
645 
646 


889 

956 

81023 


895 
963 
030 


902 
969 
037 


909 
976 
043 


916 

983 
050 


922 
990 
057 


929 
996 
064 


936 

*003 

070 


943 

*010 

077 


949 

*017 
084 


7 

7 
7 


647 
648 
649 

650 

651 
652 
653 


090 
158 
224 


097 
164 
231 


104 
171 

238 


111 

178 
245 


117 
184 
251 


124 
191 
258 


131 
198 
265 


137 
204 
271 


144 
211 
278 


151 
218 
285 


7 
7 

7 


291 


298 


305 


311 


318 


325 


331 


338 


345 


351 


7 


358 
425 
491 


365 

431 
498 


371 

438 
505 


378 
445 
511 


385 
451 
518 


391 

458 
525 


398 
465 
531 


405 
471 
538 


411 

478 
544 


418 
485 
551 


7 
7 
7 


654 
655 
656 


558 
624 
690 


564 
631 
697 


571 
637 
704 


578 
644 
710 


584 
651 
717 


591 

657 
723 


598 
664 
730 


604 
671 
737 


611 
677 
743 


617 
684 
750 


7 
7 
7 


657 
658 
659 

660 

661 

662 
663 


757 
823 
889 


763 

829 
895 


770 
836 
902 


776 
842 
908 


783 
849 
915 


790 
856 
921 


796 

862 
928 


803 
869 
935 


809 
875 
941 


816 

882 
948 


7 
7 
7 


954 


961 


968 


974 


981 


987 


994 


*000 


*007 


*014 


7 


82 020 
086 
151 


027 
092 
158 


033 
099 
164 


040 
105 
171 


046 
112 

178 


053 
119 
184 


060 
125 
191 


066 
132 
197 


073 

138 
204 


079 
145 
210 


7 
7 
7 


664 
665 
666 


217 
282 
347 ■ 


223 

289 
354 


230 
295 
360 


236 
302 
367 


243 
308 
373 


249 
315 

380 


256 
321 
387 


263 
328 
393 


269 
334 
400 


276 
341 
406 


7 
7 
7 


[667 

668 
669 


413 

478 
543 


419 

484 
549 


426 
491 
556 

2 


432 
497 
562 

3 


439 
504 
569 


445 
510 
575 


452 
517 

582 

6 


458 
523 

588 


465 
530 
595 


471 
536 
601 


7 

4 

7 


N 





1 


4 


5 


7 


8 


9 


D 



64 PLANE TRIGONOMETRY [chap, iv 

By (a), the mantissa of log 6425 = 80787 

the mantissa of log 6426 = 80794 

difference for 1 = 7 

Now the mantissa of log 6425.4 is evidently larger than 80787 and 
less than 80794, and since 6425.4 lies T 4 o the way between 6425 and 
6426, we will assume that the mantissa of log 6425.4 lies T 4 o the way 
between 80787 and 80794. We must therefore increase the smaller 
of the two mantissas by T % of the difference between the two, that is, 
by A of 7. 

T 4 o of 7 = 2.8 or 3 to the nearest integer, 

and the mantissa of log 6425.4 = 80787 + 3 = 80790. 

Hence 

log 6425.4 = 3.80790. 

We have here assumed the principle of proportional parts for loga- 
rithms of numbers, namely, that for small changes in the number, the 
change in the logarithm is proportional to the change in the number. 

Example 4. Find the logarithm of 6.5487. 

The characteristic of log 6.5487 = o. The mantissa of log 6.5487 
is the same as the mantissa of log 6548.7. 

mantissa log 6548 = 81 611 

mantissa log 6549 = 816 17 
difference for 1 = 6 

difference for 0.7 = 0.7 of 6 = 4.2 or 4 to the nearest integer. 
Hence mantissa log 6548.7 = 81611 + 4 = 81615, 
and log 6.5487 = 0.81615. 

Example 5. Find log 0.000635945. 

The characteristic of log 0.000635945 = — 4 or 6 — 10. 

The mantissa of log 0.000635945 is the same as the mantissa of log 

635945- 

mantissa log 6359 = 80339 

mantissa log 6360 = 80346 
difference for 1 = 7 

difference for 0.45 = 0.45 of 7 = 3.15 or 3 to the nearest integer. 
Hence mantissa log 6359.45 = 80339 + 3 = 80342, 

and log 6359.45 = 6.80342- 10. 



31] LOGARITHMS 65 

Example 6. Find the cologarithm of 65.021. 
By definition, Art. 26, the cologarithm of a number is the loga- 
rithm of the reciprocal of that number, hence 

colog 65.021 = log 1 — log 65.021. 

log 1 = o = 10 — 10 

log 65.021 = 1. 81306 

colog 65.021= 8.18694* — 10. 

31. To Find the Number Corresponding to a Given Logarithm. 

(a) When the given mantissa can be found in the table. The char- 
acteristic is used only to determine the position of the decimal point 
after the number has been found. Find the given mantissa in the 
table. The first three figures of the number sought are found in 
the same line with the mantissa, in the column on the left, and the 
fourth figure is found at the top of the column containing the given 
mantissa. 

Example i. Find the number whose logarithm is 5.82158. 

Find the given mantissa 82158 in the table (see specimen page, p. 63). 
The number in the left-hand column and in the same line with the given 
mantissa is 663, and the number at the top of the column containing 
the mantissa 82158 is 1, hence the significant figures of the required 
number are 6631. The characteristic is 5, hence the integral part 
of the required number has 6 places, that is, the required number 
is 663100. 

Example 2. Find the number whose logarithm is 8.81043 — 10. 

Corresponding to the mantissa 81043 we nn d in the table the num- 
ber 6463. The characteristic is 8 — 10 or — 2, hence by the rule 
for the characteristic the required number is a decimal fraction with 
one cipher preceding the first significant figure. 

Therefore the required number is 0.06463. 

(b). When the given mantissa cannot be found in the table, two other 
consecutive mantissas can always be found in the table, one of which 
is a little smaller and the other a little larger than the given man- 
tissa. The four figures corresponding to the smaller of these man- 
tissas will be the first four figures of the required number; the fifth 

* The subtraction is performed from left to right by subtracting each figure 
from 9 except the last one, which is subtracted from 10, thus: 1 from 9 = 8, 8 from 
9 = 1, 1 from 9 = 8, 3 from 9 = 6, o from 9 = 9, 6 from 10 = 4. 



66 PLANE TRIGONOMETRY [chap, rv 

figure, and sometimes the sixth,* can then be found by interpola- 
tion from the principle of proportional parts. 

Example 3. log N = 1.80395, to nn d N. 

The table does not contain the mantissa 80395, but it contains 
the two consecutive mantissas 80393 and 80400, one of which is 
smaller and the other larger than the given mantissa. The num- 
bers corresponding to these mantissas are 6367 and 6368 respectively. 

mantissa log 6367 = 80393 
mantissa log 6368 = 80400 

difference for 1 = 7 

Since the given mantissa lies between 80393 an d 80400, we infer that 
the number corresponding to the given mantissa lies between 6367 
and 6368; let it be denoted by 6367 + x, we then have 

mantissa log 6367 = 80393 
mantissa log 6367 + x = 80395 

difference for x = 2 
and the principle of proportional parts gives 

1:7 = ^:2, that is, x = y t 

and the number corresponding to the given mantissa is 63671. It 
still remains to determine the decimal point. The characteristic is 
1, hence the required number is 

N = 63. 67^ = 63.673 to five figures. 

Exercise 17 

(In this exercise the specimen page of logarithms may be used.) 

1. Find the logarithms of the following numbers: 6315, 632.5 
6.454, 0.0655, 0.0065. 

Ans. 3.80037, 2.80106, 0.80983, 8.81624 — 10, 7.81291 — 10. 

2. Find the cologarithms of each of the numbers in 1. 

Ans. 6.19963 — 10, 7.19894— 10, 9.19017— 10, 1. 18376, 2.18709. 

* In the first part of the table, the sixth significant figure of a number may be 
found by interpolation. In the latter part of the table, where the difference in 
the mantissas corresponding to a difference of i in the numbers is much smaller 
than in the first part of the table, the sixth figure of the number obtained by the 
principle of proportional parts cannot be depended on. 



32] LOGARITHMS 67 

i 

3. Find log 63.454, log 65.061, log 6.6095, log 0.0064159. 

Ans. 1.80246^1^13^^0.82017, 7.80725 — 10. 

4. Find the numbers whose logarithms are: 1.80277, 2.80584, 
0.81003, 9.81351 — 10, 8.80017 — 10, 3.81184. 

Ans. 63.50, 639.5, 6 -457> 0-6509, 0.06312, 6484. 

5. Find the numbers whose logarithms are: 1.80958, 2.81922, 
•0.81006, 9.80002 — 10, 8.80022 — 10, 3.82506. 

Ans. 64.503, 659.51, 6.4574, 0.63099, 0.063127, 6684.3. 

6. log 63.275 = x, log y = 1. 81864; find x and y. 

Ans. x = 1. 80124, y = 65.863. 

7. Without multiplying or dividing the numbers, find the loga- 
rithms of 66.027 X 0.65034, 6301 -T- 6.454, (6535.4)2, V63.275. 

Ans. 1.63287, 2.98958, 7.63055, 0.90062.. 



8. N = V64550; find log N and then N. S 

Ans. logiV = 0.80165, N = 6.^^,6. 

9. Find (6.3096) 5 by means of logarithms. Ans. iopoo. 

10. Find the logarithms of sin 40 , cos 48 30', cot 8° 50'. 

Ans. 9.80808 — 10, 9.82125 — 10, 0.80854. 
(Suggestion. First find the natural functions of the given] angles 
and then find the logarithms of the resulting numbers.) 

11. Find the first three significant figures and the number of 
figures in the integral part of the twenty-fifth power of 6.2. 

Ans. 645, 20 places. 

32. Directions for the Use of Logarithms. The following 
directions will aid the student in an intelligent use of logarithmic 
tables. 

(a) In finding the logarithm of a given number, or in finding the 
number corresponding to a given logarithm, the interpolation should 
be performed mentally and only the complete result set down in 
writing. 

(b) In writing down the cologarithm of a number, the subtraction 
from 10 should be performed mentally and from left to right. 

(c) The results obtained by logarithms are approximations only. 
By neglecting the sixth and following significant figures of a number 
the inaccuracy introduced can never exceed one-half a unit in the 



68 PLANE TRIGONOMETRY [chap, iv 

fifth place, that is, the error cannot exceed the 200 part of 1 per 
cent. 

(d) When the sixth figure of a number is 5, and we wish to retain 
only five significant figures, it is immaterial whether we increase 
the fifth figure by 1 or leave it unchanged. In case we increase 
the fifth figure by 1 the resulting number will be too large by half a 
unit in the fifth place; if we leave the fifth figure unchanged, the 
number will be too small by half a unit. In order to cause the 
inaccuracies arising from this source to offset one another, it is cus- 
tomary, on dropping a final 5, to increase the preceding figure by 1 
when it is odd, but to leave it unchanged when it is even. 

Thus, 0.154755 becomes 0.15476, 

but 0.154745 becomes 0.15474; 

one-half of 4.23453 becomes 2. 11 726, 

but one-half of 4.23463 becomes 2.1 1732, 

when the results are abridged to five places. 

(e) Sometimes the nature of the problem is such that a four-place 
table would give all the accuracy required. In that case the fifth 
figure of the mantissa in the table may be omitted and the fourth 
figure increased by 1 if the omitted figure exceeds 5. If the final 
figure of the mantissa given in the table is marked with a stroke, 
thus 5, on omitting it the preceding figure is left unchanged; but if 
the final 5 is unmarked, the preceding figure is increased by 1. The 
reason for this is that the final 5 of a mantissa is itself the result of 
approximation, that is, it is either in defect (5 plus something less 
than J), or it is in excess (4 plus something greater than |), and the 
latter case is distinguished from the former by printing a stroke 
over the 5. 

Thus from the table, 

log 2.078 = 0.31765 = 0.3176, 

but log 2.079 = °-3!785 = 0.3179, 

when abridged to four places. 

(/) Every logarithm consists of two parts, — the mantissa, which 
is always positive, and the characteristic, which is always integral 
(or o), but maybe negative as well as positive. When the character- 
istic is negative, it is customary to change the form of the logarithm 
by adding and subtracting 10. 



32] LOGARITHMS 69 

Thus, log 0.4562 = — 1 + 0.65916 is written 9.65916 — 10, 

log 0.0032 = — 3 + 0.50515 is written 7.50515 — 10. 

This is done in part to avoid mistakes which might arise from con- 
fusing the positive and the negative parts of logarithms. 

For similar reasons, if a logarithm whose characteristic is negative 
is to be divided by 2, as in extracting a square root, we first modify 
its form by adding and subtracting 20; if the logarithm is to be 
divided by 3, we first add and subtract 30; and generally, if the 
logarithm whose characteristic is negative is to be divided by n, 
we first modify its form by adding and subtracting ion. 

Thus, log 0.03254 = — 2 + 0.51242 = 8.51242 — 10, 

\ of log 0.03254 = \ of (18.51242 — 20) = 9.25621 — 10, 
\ of log 0.03254 = i of (28.51242 - 30) = 9.50414 - 10, 
J of log 0.03254 = J of (38.51242 — 40) = 9.62811 — 10, 
i of log 0.03254 = J of (48.51242 — 50) = 9.70248 — 10, etc. 
(g) The difference between two consecutive mantissas is called the 
tabular difference and is printed under D in the last column on each 
page of logarithms. At the bottom of each of the first three pages 
of logarithms the tabular differences which occur on the page are 
multiplied by each of the nine digits expressed as tenths. The re- 
sulting tables, known as tables of proportional parts, are used as an 
aid in interpolation. 

Example i. Find by logarithms the product of 37.543 by 0.85734. 
Solution. Denote the product by x. 

From table I, log 37-543 = 1-57453 

log o-85734 = 9-933!5 ~ 1 
By Art. 25, (a), log * = 1.50768 

By Art. 31, and the table, x = 32.187. 

Example 2. Find by logarithms the quotient of 6.3725 divided 
by 82.756. 

Solution. Denote the quotient by x. 

From the table, log 6.3725 = 0.80431 

By Art 26, (c), and the table, colog 82.756 = 8.08220 — 10 

log x = 8.88651 — 10 
From the table, x = 0.077003. 



70 PLANE TRIGONOMETRY [chap, iv 

Example 3. Find the square root of 0.89355. 

Solution. log 0.89355 = 9.95112 — 10 

= 19.95112 — 20 

Dividing by 2, log ^ 0.89355 = 9-97556 - 10 

Hence ^0.89355 = 0.94528. 

Example 4. Find x, if x 3 = (0.5824) 2 . 

•Solution. log 0.5824 = 9.76522 — 10 

Multiplying by 2, log x 3 = 19.53044 — 20 

= 29.53044 - 30 
Dividing by 3, logx = 9.84348 — 10 

x = 0.69740. 

-P, 8.35 X (62X) 2 XV5.673 r 1 

Example 5. x = — QSi — 0/ ^ - La - ; find x. 

(i.256) 3 X 623.7 XV5- 736 

Solution. By the rules of Art. 25, 

logx = log 8.35 + 2 log 62.5 + i log 5.673 

+ 3 colog 1.256 + colog 623.7 + i colog 5.736. 
log 8.35 
log 62.5 = 1.79588 

2 log 62.5 = 2 X I.79588 

log 5-673 = 0.75381 

J log 5.673 = ix 0.75381 

colog 1.256 = 9.90101 — 10 
3 colog 1.256 = 3 X (9.90101 — 10) 

colog 623.7 

colog 5.736 = 9.24139 - 10 
i colog 5.736 = iX (29.24139 - 30) 





— 


0.92169 






= 


3-59176 






= 


0.37690 






= 


9.70303 - 


10 




= 


7.20502 — 


10 




X = 


9-74713 - 


10 


log 


31-54553 - 


3° 




= 


1-54553 






X = 


35."8. 





Exercise 18 
Solve by logarithms: 

1. 3.784 X 7.843. Ans. 29.678. 

2. 67.845 X 0.03457. 

3. 0.67375 + 3-468. Ans. 0.19428. 

4. 92.57 ■*■ 1-3785- 



32] LOGARITHMS 

5. (68 4 .7) 2 X (0.03873) 3 . 

6. (0.8003) 3 -i- V5.73. 

7. V3284.5. 

8. (0.12562) 10 . 



10. 



11. 



12. 



1.56 X 37-8 X Vo.0753 . 

67-574 
600.45 X 1.0025 

(37-5) 2 X V6^T 





7i 


Ans. 


27.236. 


Am 


"• 5-°5- 


Ans. 


0.23946. 



4 /3.1416 X 6 X 27.3 A 

V/-^— - — — *- jl - -4»s. 4.21J 

V i.oc6X 2.738 X 10 



>56 X 2.738 X 10 
V (27.3)2 X 23.7 



V(28.92) 3 X 16.5 

13. The area of a triangle is given by the formula 

A = \ls (s — a)(s — b)(s — c), 

where a, b, c represent the three sides of the triangle and s half their 
sum. Find A when a = 617.34, b = 345.65, c = 467.75. 

Ans, 80127. 

14. In Art. 21 it was shown that 

, A 4 lc— b 

tan — = \ > 

2 V c + b 

where b is a side of a right triangle, c the hypotenuse, and A the angle 
between b and c. Find A when c = 325.76, b = 324.13. 

Ans. 5 44'. 

15. Solve the equation 2 X = 3.573.* Ans. 1.8371. 

16. Find x in (3.1416)* = 9.8697. 

17. Solve the equation 

3 2x — 12 X 3 X + 11 = o. Ans. x = o, or 2.1827. 

18. Solve 

X _x 

e2 + e 2 = 4, where e = 2.7183. 

Ans. x = d= 2.6339. 

* Such an equation as this, in which the unknown quantity appears as an 
exponent, is called an exponential equation. It is solved by first taking logarithms 
of both sides of the equation, thus: 

zlog2 = log 3.573, 
that is, 0.30103"* = 0.55303, 

from which x = 1.8371. 



72 PLANE TRIGONOMETRY [chap, iv 

33. Application of Logarithms. The solution of triangles, which 
furnishes the most important application of logarithms, will be 
fully considered later on. At present we give a few miscellaneous 
problems which are especially adapted to solution by logarithms. 



Exercise 19 

Many of the following list of problems require for their solution 
the compound interest formula, Problem 10, Exercise 16. 

1. Find the amount on $100 for 100 years at 4 % compound 
interest, (log 1.04 = 0.0170333.) Ans. $5050.4. 

2. In what time will a sum of money double itself at 8 % com- 
pound interest ? Ans. 9.007 yrs. 

3. A man bequeaths $500, which is to accumulate at compound 
interest until the interest for one year at 5 % will amount to at least 
$300, after which the yearly interest is to be awarded as a scholar- 
ship. How many years must elapse before the scholarship becomes 
available, assuming that the original bequest is made to earn 5 % 
compound interest ? Ans. 51 yrs. 

4. At what rate of interest must the bequest in Problem -3 be 
invested in order that the scholarship may become available in 
40 yrs. ? Ans. 6.4 %. 

5. In 1624 the Dutch bought Manhattan Island from the Indians 
for about $24. Suppose that the Indians had put their money out 
at compound interest at 7 % and had added the interest to the prin- 
cipal each year, how large would be the accumulated amount in 
1 9 10 ? (From White's Scrap Book of Mathematics.) 

Ans. In round numbers $6,000,000,000. The actual valuation of 
Manhattan and Bronx real and personal property in 1908 was 
$5,235,399,980. 

6. The population of the state of Washington in 1890 was 349,400 
and in 1900 it was 518,100. What was the average yearly rate of 
increase ? Assuming the rate of increase to remain the same, what 
should be the population in 1910 ? Ans. 4 %; 767,970, nearly. 

7. The founder of a new faith makes one convert each year, and 
each new convert makes another convert each year, and so on. 



34] LOGARITHMS 73 

How long would it require to convert the whole earth to the new faith 
assuming that the population of the world is 1,500,000,000 ? 

Ans. Between 30 and 31 yrs. 
8. The combined wealth of the United States and Europe was 
estimated (1908) to amount to about $450,000,000 000. Let us 
assume that the entire wealth of the world amounts to $io 12 . How 
long would it take $1 put out at compound interest at 3 % to equal 
or exceed this amount ? Ans. 935 yrs. 

34. To Compute a Table of Common Logarithms. 

A table of logarithms may be computed from the successive 
square-roots of 10 and multiplications. 

io^=Vio= 3.16228, (1) 

10 * =V / io2 = V3. 16228 = 1.77828, (2) 

I ol = V / I0 i=\/ 1 . 77828 = I-3335 2 . (3) 

io w = \ / io* = v / i.33352 = 1. 15478. (4) 

By definition, the common logarithm of a number is the power to 
which 10 must be raised to produce that number, hence from 

10* = 1. 15478, iog i-i548*= tV = 0.0625; 

IO T 2 6 = IQ8 = I.33352,. log 1.3335 = l\ = O.I250; 

I. 15478 = I-5399 2 , 

log 1-5399 = A = 0.1875; 

1. 15478 = 1.77828, 

log 1.7783 = T \ = 0.2500. 
Check. io T s = 10* = 1.77828 by (2). 
IO T6 = io i6 + t^ = IO T6 . 1Q i$ = 1.77828 X 1. 15478 = 2.05352, 

!og 2.0535 = -nr = 0.3125; 
io 16 = io 16 ^ = io 16 • io 115 = 2.05352 X 1. 15478 = 2.37136, 

log 2.3714 = T 6 6- = O.3750; 
IO T6 — IO T6 + TS — IO T6 . IO T^ = 2.37136 X 1. 15478 = 2.7384O, 

log 2.7384 = T V = 0.4375; 

g 7 _l 1 7 1 

IO T6 = joTe+TS = IO T6 . IO TS — ■ 2.7384O X 1. 15478 = 3.16225, 

log 3.1623 = T \ = 0.5000. 
Check. io T 6 = io* = 3.16228 by (1). 
IO T6 = io t 8 6 + t5- = IO T6 . io 1 * = 3.16228 X 1. 15478 = 3.65174, 

log 3.6517 = T \ = 0.5625; 

* Only 5 figures are carried in order to secure accuracy in the last figure of all 
the numbers in the list. 



3 — 2 _+ 1 -3— 

io 16 = io 16 w = io 16 • 10^= 1.33352 X 1. 15478 = 1.53992, 

IO T 4 6 = io^ +3 ^ = IQTe • IO 1 * = 1.53992 X 1. 15478 = I.77828, 



74 PLANE TRIGONOMETRY [chap, iv 

I0 i6 = io i^ + ^ = io 1 ^ • io 1 ^ = 3.65174 X 1. 15478 = 4.21696, 

log 4.2170 = if = 0.6250; 

IO T6 = IOl6 + T& = IO T6 . IO W = 4.21696 X 1. 15478 = 4.86966, 

log 4.8697 = H= 0.6875; 

IO !6 = IO Ti + T<5 = IO !6 . IO^ = 4.86966 X I- 15478 = 5.62339, 

log 5.6234 = if = 0.7500. 

12 3 1 1 

Check. io T ^ = 10* = 10^ • io 4 = 3.16228 X 1.77828 = 5.62342 by 
(1) and (2). 

13 12.1-1 12 1 

I0 T6 = IO T6 + TS = IO T6 . IO T^ = 5.62342 X 1. 15478 = 6.49382, 

log 6.4938 = it = 0.8125; 

IO T6 = IO TF + T^ = IO T6 . IO W = 6.49382 X 1. 15478 = 7.49892, 

log 7.4989 = if = 0.8750; 

IO l6 = I0 16 + TF = IO T6 . IO T5 = 7.49892 X I- 15478 = 8.65960, 

log 8.6596 = if = 0.9375; 

IO tI = I0 16 + T^ = IO T6 . IO TO = 8.65960 X 1. 15478 = 9.99993, 

log IO.OOO = if = 1. OOOO. 

Check. io T e = io 1 = 10. 

Each check is on the four lines just preceding, and in each case 
there is exact agreement to five figures between the results of the 
check and the result of the line just preceding. The discrepancy in 
the sixth figure, where it occurs, is due to the fact that all our re- 
sults are approximations only. 

We have thus computed the numbers whose mantissas are dis- 
tributed between o and 1 at intervals of T V- 

Similarly, with the aid of 



I0 32 = V IO & = Vi. 15478 = 1.07468, 



io* 1 * = v / I0 32 = V1.07486 = 1.03663, etc., 

longer lists of numbers could be computed whose mantissas differ 
by -jVj F4> e tc. By continuing this process the interval between suc- 
cessive mantissas can be made small at will. When the difference 
between successive mantissas has been made sufficiently small, the 
mantissa corresponding to any number, intermediate to two num- 
bers in the list already found, may be found by interpolation. If 



35] LOGARITHMS 75 

now the numbers on the left are selected at equal intervals and tabu- 
lated, together with their mantissas, we shall have a table of common 
logarithms.* 

The characteristics need not be tabulated, for they can always be 
determined from memory by the two rules of the preceding article. 
In most of the tables the decimal points are also omitted; for in- 
stance, we find in most tables corresponding to 

the number the mantissa 

4753 67697. 

meaning that .67697 is the fractional part of the logarithm of any 
number whose significant figures are 4753. 

The method just explained, simple as it is, is not the method by 
which the existing tables of logarithms have actually been calcu- 
lated. We shall learn later that other methods exist by means of 
which logarithms may be calculated with much greater ease and 
speed. 

35. Relation between log a N and log 6 N. We will now show that 
the logarithms of the same numbers to two different bases are pro- 
portional, so that when the logarithm of a number to a given 
base, as the base 10, is known, the logarithm of the same number 
to any other base may be obtained from it by multiplying the known 
logarithm by a certain constant. It is therefore unnecessary to 
actually construct logarithmic tables to more than one base. 

To prove that 

log^Y = fi loga^Y, where a is the constant . 

logafr 

Proof. Let log&iV = x, from which b x = N, (1) 

and log a A T = y, from which a y = N. (2) 

From (1) and (2) 

b* = a«. (3) 

Now let 

b = a c , (4) 

* The student will observe that in the original arrangement the numbers on 
the right (mantissas) are at equal intervals. In this form the table is known as 
a table of antilogarithms. A table of antilogarithms would serve the main purposes 
of computation as well as a table of logarithms. Tables of antilogarithms have 
been published and are used by some computers. 



76 PLANE TRIGONOMETRY [chap, iv 

and substitute (4) in (3), then 

(a c ) x = a cx = a y , 
from which ex = y, 

or x = 2 (5) 



where from (4), c = log a b. 

Putting in (5) for x and y their values from (1) and (2), and M for- 

c 

we have 

\og h N= ^\og a N y (6) 

where 

_ 1 

l0g a b 

(6) may also be written 

l 0g6 ^=M. (7) 

log a & 

If in (7) we put N = a, we obtain (since log a a = 1) 

log b a= — ^— , 
logJ> 

that is, 

/c^ba and log a b are reciprocals. 

The constant multiplier pi is called the modulus of the system of 
logarithms whose base is b with reference to the system whose base 
is a. 

36. Natural or Hyperbolic Logarithms. Theoretically any posi- 
tive number different from 1 may serve as the base of a system of 
logarithms, but in practice only two systems are used. The first is 
the common system, used exclusively in numerical computations; the 
other is known as the system of natural or hyperbolic logarithms, 
which is used extensively in theoretical investigations. 

The base of the natural system * is an incommensurable number, 

* Natural or hyperbolic logarithms are known also as Napierian logarithms in 
honor of John Napier (1550-1617), though this name is a misnomer, since neither 
Napier nor any of his contemporaries had any conception of the number e or the 
system of logarithms which has e for its base. Napier's base is the number 

0.367879, which happens to be nearly equal to - . The discovery of natural loga- 
rithms as well as the name is due to Nicolas Mercator (1620-1687). 



36] LOGARITHMS 77 

known as e, the twin sister of the number ir, which like w has many 
remarkable properties, e is defined as the limit which (1 + -J ap- 
proaches, as x increases indefinitely. It will be shown later that this 
limit is equivalent to the series 

i + i + *+"^- + _JL-1 + - + 

2 2-3 2-3-4 2-3-4-5 

from which the approximate value of e is readily found to be 

e = 2.71828 • • • . 

By the preceding article we find the modulus of the natural system 
of logarithms with reference to the common system is 

1 1 
fi e = = = 2.30259 • • • . 

log 2.71828 0.434294 

Exercise 20 

1. Use the results of Art. 34 to compute to four places the number 
whose common logarithm is J^ = 0.15625. Ans. 1.4330. 

2. By a method similar to that of Art. 34 compute to four places 
the number whose common logarithm is J; also to four places the 
number whose common logarithm is |. Ans. 2.1544, 1.2913. 

3. Similarly compute the number whose common logarithm is ^. 

Ans. 1.4678. 

4. Given log i 2 = 0.30103; compute log 2 10. Ans. 3.3219. 

5. Show that logio N = -^ — = 0.434294 X logJV. 

log e io 

6. Find log e 100, log e 1000, log e 0.01, log e 2. 

Ans. 4-60518, 6.90777, - 4-60518, 0.69315. 

7. Use the results of Art. 34 to compute log e 3.65174. 

Ans. 1. 29521. 

8. Given logi 3; nn d logg 10, log 2 7 1000. 

Ans. , . 

2 log™ 3 logio 3 

9. Show how to obtain: 

logarithms to the base 8 from logarithms to the base 2 ; 
logarithms to the base 3 from logarithms to the base 9. 

Ans. Divide by 3; multiply by 2. 



78 PLANE TRIGONOMETRY [chap, iv 

io. Prove that 

\og b a • \og c b • log a c = i. 

ii. Given log 4 = 0.60206, log 9 = 0.95424; find log 6. 

Ans. log 6 = 0.77815. 

37. Tables of Logarithmic Trigonometric Functions. In the 

solution of triangles and elsewhere we constantly encounter expres- 
sions involving trigonometric functions. Suppose it were required to 
compute the value of 

_ 325.6 sin 23 45' tan 18 24 ' 
cos 37 30' 

It is plain that we might first find the values of the several trigono- 
metric functions from the table of natural functions; thus, — 

sin 23°45' = 0.4027, tan 18 24' = 0.3327, cos37°3o / = 0.7934. 

With these values the expression above written becomes 

x = 3 25-6X0.4027 x 0.3327 m 
o.7934 
Solving by logarithms we have 

log 325.6 = 2.51268 

log 0.4027 = log sin 23°45' = 9.60498 — 10 

log 0.3327 = log tan 18 24' = 9.52205 — 10 

colog 0.7934 = colog cos 37 30' = 0.1005 1 

log x = 1.74022 
x = 54-9 8 3- 

It will be observed that in order to obtain log sin 23 45' we con- 
sulted two different tables: first, the table of natural sines, to find 
sin 23°45 r = 0.4027; second, the table of logarithms, to find log 0.4027 
= 9.60498 — 10. To avoid the necessity of referring to two tables, 
the logarithms of the sines, cosines, tangents and cotangents have 
been separately calculated and arranged in a new table, known as 
the table of logarithmic trigonometric functions *. The table con- 
tains the values of the logarithms of the sines, cosines, tangents 
and cotangents of angles between o° and 90 . 

We know that the sine and cosine of every angle and the tangent 
of every angle less than 45 , as well as the cotangent of any angle be- 

* The choice of the term is unfortunate. Trigonometric logarithms would be 
much better. It is of interest to know that Napier's tables, the first tables ever 
published, were tables of logarithmic trigonometric functions. 



38] LOGARITHMS 



79 



tween 45 and 90 , is less than unity, consequently the characteristics 
of the logarithms of these functions will be negative. To avoid 
negative characteristics, each negative characteristic has been replaced 
by a positive characteristic by adding 10. For example, while the 
true value of 

log sin 23 45' = — 1 + 0.60498 = 9.60498 — 10, 

the table gives 

log sin 23 45' = 9.60498. 

This latter value is called the tabular logarithmic sine, hence 

To find the true value of a logarithmic sine the corresponding tabular 
logarithmic sine must be diminished by 10. 

Since the secant is the reciprocal of the cosine, and the cosecant 
the reciprocal of the sine, we have 

The logarithmic secant or cosecant may be obtained by taking the 
cologarithm of the cosine or sine respectively, that is, by subtract- 
ing the tabular logarithmic cosine or sine from 10. 

38. To Find the Logarithmic Trigonometric Functions of an 
Angle less than 90°. 

(a) When the angle is less than 45 , we find the number of degrees 
at the top of the page and the number of minutes in the left-hand 
column. The values of the log sin, log tan, log cot, log cos, as indi- 
cated at the head of the column, are then found in the same line as 
the minutes. 

If seconds are given, they may be reduced to a fractional part of a 
minute, and the value of the logarithmic function may be found by 
interpolation, just as was done in finding logarithms of numbers. 

Example i. Find the log sin, log cos, log tan and log cot of 

II°2l'. 

Remembering that certain of the characteristics as given in the 
table are too large by 10, we find from table II 

log sin ii° 21' = 9.29403 — 10, 
log tan n° 21' = 9.30261 — 10, 
log cot ii° 21' = 0.69739, 
log cos n° 21' = 9.99142 — 10. 



8o PLANE TRIGONOMETRY [chap, iv 

Example 2. Find log sin 15 24/ 36". 
From the table we find, 

log sin 15 24/ = 9.42416 

log sin 15 25' = 9.42461 

difference for i' = 45 

36" = o.6', difference for o.6' = 0.6 X 45 = 27, 

hence log sin 15 24' 36" = 9.42416 + 27 = 9.42443. 

Example 3. To find log cos 27°36 / 4o // . 
From the table we find 

log cos 27°36' = 9.94753 
log cos 27°37 / = 9.94747 
difference for 1' = 6 

40"= o.6§', difference for o.6f = 4, 

hence log cos 2 f 36' 40" = 9.94753 ~ 4 = 9-94749- 

Observe that in this example the difference was subtracted from the 
mantissa of the logarithm of the smaller angle, because the logarithm 
of the cosine decreases as the angle increases. 

(b) When the angle is more than 45 and less than <?o°, we find the 
degrees at the bottom of the page and the minutes in the right- 
hand column. The values of the log sine, log tangent, log cotangent 
and log cosine as indicated at the foot of the column are then found 
in the same line as the minutes. 

Example 4. Find log tan 6i° io' 27". 
From the table we find 

log tan 6i° 10' = 0.25923 
log tan 6i° n' = 0.25953 
difference for i f = 30 

27" = 0.4^, difference for 0.4^ = 4! X 30 = 13.5, 

hence log tan 6i° io ; 27" = 0.25923 + 13.5 = 0.25936. 

The difference 13.5 is additive because the log tangent increases with 
the angle. 



39] LOGARITHMS 8 1 

Example 5. 



LOGARITHMS 


Find log cot 75° 5*' 15"- 




log cot 75 51' = 


9.40159 


log cot 75 52' = 


9.40106 


difference for i' = 


53 


difference for 0.25' = 


*3, 


log cot 75°5i / 15" = 


9.40146 



15" = 0-25', 

hence log cot 75 51' 15" = 9.40146 — 10. 

The difference 13 is subtracted because log cot 7 5 52' is less than 
log cot 75 51'. 

In general when interpolating: 

For sines and tangents the difference must be added to the function 
of the smaller angle, because the sine and tangent of an angle in- 
creases as the angle increases. 

For cosines and cotangents the difference must be subtracted from 
the function of the smaller angle, because the cosine and cotangent 
of the angle decreases as the angle increases. 

39. To Find the Angle Corresponding to a Given Logarithmic 
Trigonometric Function. 

When the given number can be found in the table, the number of 
degrees is found at the top or bottom of the page, according as the 
name of the function appears at the top or bottom of the column. 
The number of minutes is found in the same line with the given 
function, — in the left column if the degrees are taken from the top 
of the page, in the right column if from the bottom. 

When the given number cannot be found in the table, two con- 
secutive numbers can always be found, one of which is slightly 
smaller, the other slightly larger than the given number. The re- 
quired angle can then be found by interpolation. 

Example i. Given log sin x = 8.73997; to find x. 

The number 8.73997 is found in the first column on page 30 of the 
tables. This column has log sin written at the top and log cos at 
the bottom. Since the given number is to be a log sine, the degrees 
are taken from the top of the page and the number of minutes from 
the left-hand column in line with the number 8.73997. We thus find 
x = 3 09'. 

Example 2. Given log cos y = 8.73997; to find y. 
The given number is the same as in Example 1, but this time it 
represents a log cosine. Therefore we take for the required angle 



82 PLANE TRIGONOMETRY [chap, iv 

the number of degrees at the bottom of the page and the number 
of minutes found in the right-hand column, opposite the number in 
the table. Result, y = 86° 51'. 

It should be observed that the given mantissa 73997 appears a 
second time in the table, namely, on page 60, first column. But 
the numbers in that column have the characteristic 9, while the 
given characteristic is 8. Beginners sometimes overlook the charac- 
teristic and are thus led into mistakes. 

Example 3. Given log tan x = 0.08685; to find x. 

The number 0.08685 cannot be found in the table, but on page 66, 
third column, are found the next smaller number 0.08673 an d the 
next larger number 0.08699. The difference between these two is 
26, and to the smaller of the two corresponds the angle 5o°4i / , 

that is, 

mantissa log tan 50 41' = 08673 

but mantissa log tan x = 08685 

difference = 12. 

If we denote by d" the difference between 5o°4i' and x, the principle 
of proportional parts gives us 

d" : 12 = 60" : 26, that is, d" = if X 60" = 28", 

and the required angle is x = 50 41' 28". 

Example 4. Given log cos x = 9.87561 — 10; to find x. 

mantissa log cos 41 19' = 87568 
mantissa log cos 4i°2o / = 87557 
difference for i' or 60" = 11. 

Also mantissa log cos 41 19' = 87568 

mantissa log cos x =87561 

difference for d" — 7 

and d" : 7 = 60" : 11, d" = t 7 t X 60" = 38", 

hence the required angle is x = 41 19' 38". 

Exercise 21 
From the table of logarithmic trigonometric functions find: 

1. log sin 13 24', log cos 25 i2 r , log tan io°o2 / , log cot i7°oo / . 

Ans. 9.36502, 9.95657, 9.24779, 0.51466. 

2. log sin 72 11', log tan 46 17', log cot 65 13', log cos 67 59'. 

Ans. 9.97866, 0.01946, 9.66437, 9-573 8 9- 



4 oj LOGARITHMS 83 

3. log sin 25 12' 30", log tan 36 30' 15", log sin 70 15' 40" '. 

Ans. 9.62932, 9.86928, 9.97370. 

4. log cos 16 26' 45", log cot 9 27' 42", log cot 54 25' 09". 

Ans. 9.98186, 0.77818, 9.85456. 

5. log sin 42 15' 10", log cos 17 10' 54", log tan 51 18' 57". 

Ans. 9.82763, 9.98017, 0.09653. 

By means of the table find x, when there is given: 

6. log sin x = 9.77980, log tan x = 0.40017, log cos x = 9.79558. 

Ans. 37°o2 r , 68° 18', 5i°2i'. 

7. log sin x = 9.68931, log cos x = 9.89609, log tan x = 0.16999. 

Ans. 29° 16' 30", 38 04' 30", 55°56'i 5 ". 

8. log tan x = 0.30562, log sinx= 8.97296, log sin x = 9.97296. 

Ans. 63 40' 36", 5° 23' 30", 69 S9 ' 24". 

9. log sin x = log 2 + log sin 3 2° 10' 15" + log cos 32 io' 15". 

^4^5. # = 64 20' 30". 

10. Verify the relation 

log cos 27 io'4o" = log (cos i3°35 / 20" — sin i3 35 r 20") 
+ log (cos i3°35 ; 20" + sin i3°35 r 2o r/ ). 
(Suggestion. Use the natural function table to find the quantities 
within the parentheses.) 

11. Given tan x = 4.6525; find x without using a table of natural 
functions. 

12. Given (sin x)* = 0.253; find x. Ans. x = 7 i8'4o". 

40. Logarithmic Functions of Angles near 0° or 90°. When 

the angle is very small, say less than 2 , the logarithms of the 
sine, tangent and cotangent vary so rapidly that the principle of 
proportional, parts fails to apply with sufficient accuracy, causing 
the results obtained by interpolation to be unreliable. The same 
remark applies to the logarithms of the cosine, cotangent and tangent 
of an angle near 90 , say between 88° and 90 . In such cases tables 
may be used which give the required functions for sufficiently small 
intervals of the angle, say for each second. When such tables are 
not available the following rules may be employed. 

If the angle is less than 2 and only tenths of minutes are con- 
sideredj — 



84 PLANE TRIGONOMETRY [chap, iv 

The log sin or log tan of the angle may be found by increasing the 
logarithm of the number of minutes in the angle by 6.46373-10. 

Conversely, 

The angle corresponding to a given log sin or log tan may be found by 
diminishing the given log sin or log tan by 6.46373-10. The resulting 
number is the logarithm of the number of minutes in the required angle. 

Example i. To find log sin o° 50. f. 

log 50. 7 = 1. 70501 

6-46373 ~ J Q 
log sin 50. 7' = 8. 16874 — 10 

Example 2. Given log tan x = 8. 50724 — 10; to find x. 

log tan x = 8.50724 — 10 

6 -46373 ~ 10 
logx= 2.04351 

x = 110.5' = i°5°-5'- 
These rules are based on the theorem, which will be proved later, that 
the sine and tangent of a small angle are each approximately equal 
to the length of the arc subtended by the angle at the center of a 
circle whose radius is the unit of measure, so that for very small angles 
the measure of the arc may be substituted for either the sine or the 
tangent of the corresponding angle. Now the measure of an arc in 
terms of the radius is found by multiplying the number of minutes in 
the arc by 0.0002909, for since the semi-circumference (the arc sub- 
tending an angle of 180 = 10800') measures 3. 141 59 . . . when the 
radius is 1, each minute of arc measures 3. 141 59 -r- 10800 = 0.0002909. 
Log 0.0002909 = 6.46373 — 10, consequently if the log sin or log tan 
of a small angle is increased by this amount the result will be the loga- 
rithm of the number of minutes in the corresponding angle. 

When the angle is less than 2 and the exact number of seconds 
are considered the foregoing rule fails. In such cases a special table, 
known as the S and T table, is commonly employed. 

41. Use of the S and T Table (Table III). For small angles 
the ratio of either the sine or tangent to the length of the arc of unit 
radius is nearly constant, that is 

sin x T1 tan x • , . . 
» as well as > is nearly constant, 

x being the ratio of the length of the arc to the radius. 



41] LOGARITHMS 85 

On taking logarithms we have 

log sin x — log x, as well as log tan x — log %, is nearly constant. 

Let us put 

log sin x — log x = S, and log tan x — log x = T, 
then on transposing log x we obtain 

log sin oc = log oc + S (1) 

and 

log tan a? = log oc + T. (2) 

The values of S and T, corresponding to various values of x ex- 
pressed in seconds, have been carefully calculated and assembled in 
Table III. The given angle x is first reduced to seconds, and the 
corresponding value of S or T is then taken from the table. This 
value increased by log x gives log sin x or log tan x, as the case 
may be. 

Conversely, if log sin x or log tan x is given and x is to be found,, 
we take S or T from the table, subtract it from log sin x or log tan x v 
as the case may be, and obtain log x, from which x is found. 

The log cosine or log cotangent of an angle near 90 may be found 
from the same table by substituting for the cosine of the angle the 
sine of its complement, and for the cotangent of the angle the tan- 
gent of its complement. All this will be better understood by fol- 
lowing through the examples which are worked out below. 

The characteristics of S and T are negative, so that — 10 must be 
appended to each value taken from the table. 

Example i 

Given x = o° 56' 26"; to find log sin x. 

x = o° 56' 26" = 3386". 
Applying (1), log x = 3.52969 

From Table III, 5 = 4.68556 - 10 

log sin x = 8.21525 — 10. 

Example 2 

Given log sin x = 8.21524 — 10; to find x. 

log sin x = 8.21524 — 10 
From Table III, S = 4.68555 - 10 

By (1), log* = 3.52969 

x = 3386" = o° 56' 26". 



86 PLANE TRIGONOMETRY [chap, iv 

Example 3 

Tofindlogtani°io'5i". . 

i° 10' 51"= 4251". 
Applying (2), log 4251 = 3.62849 

From Table III, T = 4.68564 - 10 

log tan i° io' 51" = 8.31413 — 10. 

Example 4 

Given log tan x = 8.31413 — 10; to find x. 

log tan x = 8.31413 — 10 
From Table III, T = 4.68564 — 10 

Subtracting, log# = 3.62849 

x = 4251" = i° 10' 51". 

Example 5 
To find log cos 89 25' 11". 

log cos 89 25' 11" = log sin o°'34 ; 49" 
o° 34' 49" = 2089". 
Applying (1), log 2089 = 3.31994 

From Table III, S = 4.68557 - 10 

log sin o° 34' 49" = 8.00551 — 10 
log cos 89 25' 11" = 8.00551 — 10. 

Example 6 

log cos x = 8.00551 — 10; to find x. 

log cos x = log sin (90 — x) . 

log sin (90 — x) = 8.00551 — 10 
From Table III, S = 4.68557 — 10 

log (90 —.%) = 3.31994 
90 — x = 2089" = o° 34' 49" 
x = 90 - o° 34 r 49" = 89° 25' 11". 

Example 7 
To find log cot i°o' 29". 

log cot i°o' 29" = 



log tan i°o' 29" 
= — log tan i°o' 29". 



42] LOGARITHMS 87 

Then, as in Example 3, 

log tan i° o' 29" = 8.24541 — 10 
log cot i° o' 29" = 10 — 8.24541 

= 1-75459- 

Example 8 

log cot x = 1.75459; to find x. 

log cot x = — log tan x = 1.75459, 
log tan x = 8.24541 — 10. 

Then, as in Example 4, 

x = i°o' 29". 

Exercise 22 

1. Find log sin i° 20' 02", log tan o° 45' 45", log cos 88° 54' 36". 

Ans. 8.36696 — 10, 8.12414 — 10, 8.27928 — 10. 

2. Find log sin o° 50' 50", log cos 89 oi' 55", log tan i° 15' 17" '. 

3. What would have been the error in each of the functions in 2, 
if their values had been found from table II by interpolation ? 

4. Find each of log tan 88° 05' 20", log cot 89 16' 50" from Table 
III, and check your result by Table II. 

5. Given log sin x = 8.22925 — 10, log tan y = 8.43340 — 10; find 
x andy. Ans. x = o° 58' 17", y = i° 33' 14". 

42. Historical Note. With the awakening of science in the six- 
teenth century, measurement became more precise, the resulting- 
numbers more complex, and computation more and more tedious 
and time-consuming. The demand for shorter methods of com- 
putation than were then known led to the invention of loga ithms. 
It is therefore not very strange that the method of logarithms should 
have been developed independently and almost simultaneously by 
two mathematicians, John Napier, a Scotchman, and Jost Buergi, a 
German. Napier's tables were published in 16 14. Buergi's tables 
were computed before 161 1 but not published till 1620- Both Napier's 
and Buergi's tables were soon superseded by Briggs' tables. Briggs' 
tables contained fourteen place logarithms of all numbers from 1 to 
20,000 and from 90,000 to 100,000. Briggs' tables were completed 
by Adrian Vlacq (1628), who shortened Briggs' tables to ten places, 



88 PLANE TRIGONOMETRY [chap, tv 

and computed the logarithms of the remaining numbers from 20,000 
to 90,000. 

Briggs' and Vlacq's tables are substantially the same as the tables 
in use to-day, though the tables have been checked and parts of them 
recomputed many times. The most complete check was under- 
taken by the French authorities in 1784. It required the work of 
nearly one hundred mathematicians and computers for over two 
years. The resulting tables, giving fourteen place logarithms of all 
integers from 1 to 200,000, besides natural sines and logarithmic 
sines and tangents, have never been published. Two manuscript 
copies are preserved, one at the Observatory, the other at the Insti- 
tute in Paris. 



CHAPTER V 



LOGARITHMIC SOLUTION OF RIGHT TRIANGLES AND 

APPLICATIONS 

43. Logarithmic Solution of Right Triangles. In Article 20 it 
was shown how to solve right triangles by means of natural functions. 
Now we shall employ logarithms, which as a rule shortens the work. 
We shall illustrate each of the cases which may arise by an example. 



Example i. 

Given a = 316.5, 
c = 521.2. 

Solution. 

(a) To find A and B. 

sin A = - . 

c 

log sin A = log a + colog c. 
\oga= 2.50037 
colog c = 7.28300 — 10 




Required A = 37 23. 5', 

B = 52° 36.5', 
b = 414.1. 



Fig. 30. 



(b) 


To find b. 






cos A = 


- , or b = c 
c 


COS 


A. 




log b 


= log c + log 


COS 


A. 




logc 


= 2.71700 






log 


cos A 


= 9.90009 — 


10 





log b = 2.61709 
b = 414.1. 



log sin A = 9.78337 - 10 
A = 37° 23. 5 r - 
£ = 52° 36-5'- 
(c) Check. If our result for b is correct, it must satisfy the rela- 
tion a 2 -f- b 2 = c 2 , or c 2 — b 2 = (c .+ b) (c — b) = a 2 , 

from which log (c + b) + log (c — b) = 2 log a. 

c+ b = 521.2 + 414.1 = 935.3, 

c— b = 521.2 — 414.1 = 107. 1. 
log (c + b) = 2.97095 log a = 2.50037 

log (c — b) = 2.02979 2 

5.00074 5- 000 74 

Since & checks, A may be assumed to be correct also, for b was found 
from A. 

89 



9° 



PLANE TRIGONOMETRY 



[chap. V 



Example 2. 

Given a = 6.325, 
b = 7.328. 

Solution. 

(a) To find A and B. 

tan A = - . 




Required A = 40 47.9', 
B = 49 12. 1 , 
c = 9.680. 



Fig. 31. 



(b) To find c. 



a 



sin A = - , ore = 



a 



c sin A 

log c = log a + colog sin A . 
log a = 0.80106 
colog sin A = 0.18483 



log c = 0.98589 
c = 9.680. 



log tan A = log a + colog ft. 
log <z = 0.80106 
colog & = 9.13501 — 10 
log tan A = 9.93607 — 10 
A = 40 47.9'. 
B = 49 12.1'. 

(c) Check, c 2 — b 2 = a 2 , log (c + b) + log (c — 6) = 2 log a. 
c + b = 9.680 + 7.328 = 17.008, 
c — b = 9.680 — 7.328 = 2.352. 
log (c -h b) = 1.23065 log a = 0.80106 

log (c - b) = 0.37144 2 

1.60209 1. 60212 

Here there is a slight discrepancy in the check. The discrepancy is 
in the last figure only; that is, if the mantissas of the final results 
are cut down to four figures, each side is 1.6021. Agreement in the 
first four figures of the mantissas of the final logarithms of the check 
is all that can be expected when a five-place table is used. 



Example 3. 




I 


? 


Required 


Given c = 35- J 45> 
A = 25 2 4 r 


3°"- 




a 


a = 15.079, 

* = 3i-745> 

£ = 6 4 °35 , 3o 


Solution. 


A 

Fig. 32. 


(a) To find a. 








(b) To find b. 


• A a 

sin A = - . 
c 








A ft ' 

cos A = -. 
c 



log a = log c + log sin A 
logc = 1.54586 
log sin A = 9.63252 — 10 
log a = 1. 17838 
a = 15.079. 



log b = log c + log cos A . 
log c = 1.54586 
log cos A = 9.95582 — 10 
log b = 1. 50168 
ft = 31-745- 



43] 



RIGHT TRIANGLES AND APPLICATIONS 



91 



(c) Check. 

log (c + b) + log (c - b) --= 2 log a. 
c -\- b — 66.890, c — b = 3.400. 

log (<; + &) = 1.82536 
log (c-b) = 0.53148 



log a = 1. 17838 





2.35684 2.35676 


Example 4. 


# 


Given & = 25.01, 


C^-ftAo' 


Required a = 11.57, 


B = 65 10'. 


^-"^ &•= 25.01 


c = 27.56, 


Solution. 


^ Fig. 33. 


A = 24° 50' 


(a) To find a. 


(b) To find c. 


tan B = - . 


sini) = - . 


a 




c 



log a = log b + log cot £. 
log b = 1.39811 
log cot B = 9.66537 — 10 

log a = 1.06348 
a = 11.574. 



log c = log b + colog sin B. 
log & = 1.39811 
colog sin B = 0.04214 

log c = 1.44025 
c = 27.56. 



(c) Check. 



log (c + b) + log (c — 6) = 2 log a. 
c-\- b = 52.568, c — b = 2.548. 



log (c + b) = 1.72072 
log (c — &) = 0.40620 



log a = 1.06348 
2 







2.12692 


2.12696 


Example 5. 








Given c = 34.57, 
6 = 34.04. 


^1- 


1 

^^^6 = 34.04 


? Required ^4 = io° 02. 7' 
-B = 79 57.3' 




Fig. 34- 


a = 6.030. 



Solution. In this problem the given side and hypotenuse are 
so nearly equal that the method employed in Example 3 does not 
give sufficiently accurate results. We therefore use the formulas of 
Art. 21. 



92 PLANE TRIGONOMETRY [chap, v 

(a) To find A and B. (b) To find a. 

, A 4 c — b • A a 

tan — = 1/ . sm yl = 



2 V c -f- 6 c 

, , A in / t\ i i / i j.m log a = log c+ log sin A 
log tan- = - [log (c — b) + colog (c + b)\. , & > & 

22 logc = 1.53870 

log sin^4 = 9.24163 — 10 

c- b = 34.57 - 34.04 = 0.53, log a = 0.78033 

c -{- b = 34.57 + 34.04 = 68.61. a = 6.030. 

log (c — b) — 9.72428 — 10 
colog (c + b) = 8.16361 — 10 

2)17.88789 — 20 
log tan \ A — 8.94394 — 10 
\A = 5°oi' 22" 
A = io° 02' 44", or io° 2.7' to the nearest tenth minute. 
B = 79 57' 16", or 79 57.3' to the nearest tenth minute. 

(c) Check. 

log (c + b) + log (c — b) = 2 log a. 
c+b = 34.57 + 34.04 = 68.61, 
c- b = 34.57 - 34.04 = 0.53. 
log (c + b) = 1.83639 loga^ 0.78033 

log (c — b) = 9.72428 — 10 2 

1.56067 1.56066 



Exercise 23 

Solve the following problems by logarithms, computing all angles 
to the nearest second and all sides to five significant figures. It is 
expected that the student check his results when no answers are 
given. 

1. a = 168.92, c = 289.64. 

Find A = 35° 4o' 33", B = 5 4° 19' 27", & = 235.28. 

2. a = 43.148, b = 84.107. Find^4 = 27°o9 / 29", c = 94.530. 

3. b = 2.5346, c = 3-7132. Find a > A > B - 

4. a = 0.37640, b = 0.28634. Find A and B. 

5. a = 547.5, 4 = 32 15' 24". Find b = 867.10, c = 1025.4. 

6. a = 6700, B = 27 30'. Find b and c. 



44] RIGHT TRIANGLES AND APPLICATIONS 93 

7. c = 672.34, A = 35 16' 2 5 ,/ . 

Find B = 54 43' 35", a = 388.26, & = 548.90. 

8. c = 1.001, B = 45 45' 45". Find a and 6. 

9. c = 369.27, 6 = 235.64. Find A = 50 20' 54", 5 = 284.31. 

10. c = 5464-35* a = 545 2 - I 3- Find B = 3 49' 57". 

44. Number of Significant Figures. Numerical problems are of 
two kinds: 

(a) Those in which the given numbers are exact numbers. 

(b) Those in which the given numbers are approximate only. For 
example, when we say that each of the sides of a hexagon inscribed in a 
circle with unit radius is 1, and each angle 120 , 1 and 120 are exact 
numbers, that is, the sides in question are to be considered neither 
more nor less than 1, and the angles neither more nor less than 
120 . On the other hand, when we say that the side of a field meas- 
ures 631.7 feet and the angle at a corner 73 37', the numbers 631.7 
and 73 37' are mere approximations. So far as we know the exact 
length of the measured side may be any number between 631.65 
and 631.75 and the measured angle may have any value between 
73 36.5' and 73 37. 5'- 

I. When the given numbers of a problem are exact numbers, the 
results asked for can be carried out to as many significant figures as 
the number of figures in the mantissas of the logarithms used in the 
solution. In this book, where the computations are based on a five- 
place table,* lengths must be limited to five significant figures and 
angles to seconds. Even then the fifth figure and the number of 
seconds cannot always be relied upon. 

11. When the given numbers of a problem are the results of measure- 
ment, the answers need not contain more significant figures than 
the least accurate of the given parts. Thus, if one side of a triangle 
is measured to the nearest inch and another to the nearest tenth of 
an inch, the answer for the third side need only be given to the 
nearest inch. The following directions will assist the student to 
make consistent measurements and to avoid useless calculations. 

1. Distances expressed to three figures call for angles expressed 
to the nearest five minutes, and vice versa. 

* Five-place tables answer most of the demands of applied science. The in- 
struments ordinarily used by engineers read angles to the nearest minute only. 



94 



PLANE TRIGONOMETRY [chap, v 



2. Distances expressed to four figures call for angles expressed 
to the nearest tenth of a minute* and vice versa. 

3. Distances expressed to five figures call for angles expressed 
to the nearest second, and vice versa. 

4. Distances expressed to six figures call for angles expressed to 
the nearest tenth of a second, and vice versa. A six-place table must 
be used to obtain like accuracy in the answers. 

5. Distances expressed to seven figures call for angles expressed to 
the nearest hundredth of a second, and vice versa. A seven-place table 
is necessary to obtain like accuracy in the answers. 

In this connection the student should observe that, whenever a 
number is the result of measurement or other approximation, a cipher 
to the right of a decimal fraction has a distinct significance and cannot 
be dropped at will, as is customary in dealing with exact numbers. 
For example, the square root of 3 is approximately represented by 
each of the numbers 1.7, 1.73, 1.732, 1.7320, 1.73205, etc., the ap- 
proximation being closer the more figures we write; but 1.7, when 
used as an approximation for V3,_has not the same meaning as 
1.70, for the former means that v 3 has some value between 1.65 
and 1.75, while the latter means that the number represented has 
some value between 1.695 and 1.705, which is not true. Simi- 
larly the numbers 62, 62.0, 62.00, when they represent measures of 
distances or other quantities, are not equivalent. The first implies 
that the measurement has been carried out to the nearest unit, the 
second to the nearest tenth, and the third, 62.00, that the measure- 
ment has been made to the nearest hundredth of a unit. 

45. Applied Problems Involving Right Triangles. The follow- 
ing six sections deal with applied problems involving the solution of 
right triangles. The problems are grouped with reference to the 
questions dealt with, and the problems in each set are so arranged 
that the more difficult come last. It is not expected that any one 
student work every problem, but only as many as may be necessary 
to make him reasonably familiar with the method of solving right 
triangles by means of logarithms. After that an additional hour or 
two may profitably be spent in the analysis of the more difficult 
problems involving two or more right triangles. In problems where 
no answer is given the result must be checked by the student. 

* The nearest 10" is somewhat closer. 



46] RIGHT TRIANGLES AND APPLICATIONS 95 

46. Heights and Distances 

Exercise 24 

1. From a point 185 feet from the foot of a wireless telegraph 
mast, the top of the mast was found to form an angle of 52 . Find 
the height of the mast. Ans. 237 ft. 

2. A man walking . along a straight road observes a church in a 
direction making an angle of 50 with the road. After walking 
another mile, he comes to the crossroad on which the church is 
located. The roads cross at right angles. How far is the church 
from the intersection of the roads? Ans. 1.19 miles. 

3. The summit of a mountain, known to be 14,450 feet high, is 
seen at an angle of elevation of 29 15' from a camp located at an 
altitude of 6935 feet. Compute the air-line distance from the camp 
to the summit of the mountain. Ans. 2.9 miles. 

4. The ratio of the height of a roof to its span is one-fourth 
(quarter pitch), what is the inclination of the roof to the horizontal 
line? Ans. 26°34'. 

5. During a storm a tree was broken into two parts which re- 
mained connected. The broken part made an angle of 35 with the 
ground and its top reached a mark 165 feet from the foot of the tree. 
Required the height of the remaining stump and the height of the tree 
before it broke. Ans. 116 ft., 317 ft. 

6. In constructing a grand-stand, timbers 28 ft. long are to be 
inclined at an angle of 25 and supported by four uprights, one at 
each end and two at equal distances between the two ends. How 
far apart must the uprights be placed and what are their lengths, 
the shortest being 6 ft. long? 

7. A flagpole 25 ft. long stands on a building whose height is 
unknown. From a point at the same level as the foot of the build- 
ing the angles of elevation of the top and bottom of the flagpole are 
measured and are found to be 57 and 53 respectively. Required 

the height of the building. 

Ans, By natural functions, 156 ft. 

8. From the top of a tower the angle of depression of a point in 
the same horizontal plane with the base of the tower is observed to 
be 47 13'. What will be the angle of depression of the same point 
as seen from a position halfway up the tower? Ans. 28 23'. 



96 PLANE TRIGONOMETRY [chap, v 

9. A spherical balloon whose radius is 10 ft. subtends an angle of 
i°46 / , while from the same position and at the same time the angle 
of elevation of the center of the balloon is 54 . Determine the 
height of the center of the balloon. Ans. 525 ft. 

10. An observer finds that the top of a spire due south of him has 
an angle of elevation of 25 36'. He goes to a point 650 ft. east of 
his first position and now finds that the spire bears 4o°i2 / south- 
west. Find the height of the spire. 

11. It was found that the shadow of a tall factory chimney length- 
ened 85 ft. while the sun's elevation changed from 59 to 42 . Re- 
quired the height of the chimney. Ans. 167 ft. 

47. Problems for Engineers. It is suggested that the student use 
the graphic method in checking the problems in this set to which no 
answers are given. 

Exercise 25 

1. A branch railroad is to be constructed from a point A to a 
second point B which is 5.95 miles east and 9.36 miles north of the 
first. What will be the direction of the road, assuming that it fol- 
lows a straight line? Ans. N. s 2 ° 2 7 r E. 

2. To determine the width of a stream a surveyor measures a 
line AB 375 ft. long along one bank. At B he turns a right angle 
and his assistant places a stake in the line of sight at C on the oppo- 
site bank of the stream. The angle BAC measures 64°42 / . How 
wide is the stream? . Ans. 793 ft. 

3. On a map on which 1 inch represents 1000 ft., contour lines 
are drawn for differences of 100 ft. in altitude. What is the actual 
inclination of the surface represented by that portion of the map 
at which the contour lines are one-fourth inch apart ? • 

Ans. 21 48'. 

4. A bolt 2 inches in diameter has six threads to the inch. What 
is the inclination of the thread to a cross section of the bolt ? 

Ans. i°3i.2 r . 

5. A car track runs from A to B, a horizontal distance of 1275 ft. 
at an incline of 7° 45', and then from B to C a horizontal distance of 
1585 ft. C is known to be 509 ft. above A. What is the average 
inclination of the track from B to C? 



47J RIGHT TRIANGLES AND APPLICATIONS 97 

6. Two towns A and B, of which B is 25 miles northeast of A, are 
to be connected by a new road. Ten miles of the road is constructed 
from A in the direction N. 23 ° E., what must be the length and direc- 
tion of the remainder of the road, assuming that it follows a straight 
line? Ans. 16.17 miles, N. 58 23.8' E. 

7. A surveyor wishes to ascertain the distance between two in- 
accessible objects A and B. He starts from a point C in a straight 
line with A and B and measures in a direction at right angles to AB 
a distance CD equal to 500 ft. At D he measures the angles sub- 
tended by AC and BC and finds them to be 75 35' and 34 46' 
respectively. Find the distance between A and B on the supposi- 
tion, — 

(a) That C is between A and B, 

(b) That B is between A and C. 

^4?w. (a) 2292, (b) 1598. 

8. One end of a connecting rod AB, 5 ft. long, is fastened to a 
crank ^0, 1 ft. long, while the other end is fastened to a crosshead A 

which is constrained to move along AO. 
How far from the extreme position P of the 
crosshead will A be, — 

(a) When OB is perpendicular to AB? 
Fi S- 35- (b) When angle BOA = 6o°? 

Ans. (a) V26 — 4 = 1.099 ft. 
0>) i^97+J- 4= 1.324ft. 
(Suggestion. In case (b) drop a perpendicular .BC from B to ^40, 
find OC and BC, then from the triangle ABC find 4C.) 

9. Two railroad tracks intersect at an angle 
of 54 16'. They are to be connected by a curve 
AB of 100 ft. radius. Find how far from the 
intersection point of the tracks the curve begins 
and the length of the curve. 

Ans. OB = 51.25 ft. Arc AB = 94.71 ft. 

10. Two pulleys whose radii are 18 inches and 30 inches respec- 
tively are 8 ft. apart from center to center. Find the length of the 
belt connecting the pulleys, — 

(a) If the pulleys are to turn in the same direction, 

(b) If the pulleys are to turn in opposite directions. 





98 PLANE TRIGONOMETRY [chap, v 

48. Applications from Physics. 

Exercise 26 

1. The horizontal distance between the two extreme positions of 
a pendulum 39.1 inches long is 5.73 inches. Through what angle does 
it swing? Ans. 8° 24'. 

2. Two forces of 10 and 24 lbs. respectively act at right angles to 
each other. Find the resultant force, and also the angle which the 
resultant makes with the first of the two given forces. 

Ans. 26 lbs., 67 23'. 

3. What force is necessary to roll a barrel weighing 500 lbs. onto a 
platform 6 ft. high along an inclined ladder 12 ft. long? 

Ans. 250 lbs. 

4. A ball weighing 300 lbs. rests on a smooth plane inclined at an 
angle of 12 30' to the horizontal. What force is necessary to keep 
the ball from rolling down the plane, 

(a) If the force acts parallel to the inclined plane ? 

(b) If the force acts in a horizontal direction ? 

Ans. (a) 64.93 lbs., (b) 66.51 lbs. 

5. A block of wood rests on an adjustable inclined plane. As the 
inclination of the plane reaches 29 37' the block begins to slide. 
Find the coefficient of friction.* 

6. An automobile moving at the rate of 45 miles per hour is over- 
taken by a shower. As seen from the automobile the raindrops 
seem to come down at an angle of 30 with the vertical. Find the 
velocity of the raindrops, assuming that their actual direction is 
vertical. Ans. 37.5 ft. per sec. 

7. Through what angle must a fir log 30 ft. long and 54 inches in 
diameter, standing on end, be tilted before it begins to fall? The log 
is assumed to be cylindrical in shape. Ans. 7 33'. 

8. According to Wollaston the intensity of sunlight is equal to 
61,000 standard candles acting at a distance of 1 meter. What is 
the intensity of sunlight striking a surface at an angle of 31 08' 27"? 

Ans. 52210 c.p. 

9. The fans of a windmill are inclined 25 to the plane of the wheel 
which is at right angles to the direction from which the wind 

* The coefficient of friction is equal to the tangent of the angle of inclination 
of the plane on which the block rests. 



4 8] 



RIGHT TRIANGLES AND APPLICATIONS 



k 



99 




blows. What fraction of the wind's force is effective in turning the 
wheel? Ans. 0.383. 

10. A weight of 437 lbs. is suspended and pushed 17 30' out of the 
vertical by a horizontal force. Required the horizontal force neces- 
sary to hold the body in this position. 

11. What is the displacement CM, of a ray of 
light AB in passing through a glass plate PQ, 
0.215 inches thick, at an angle of 55 47' with 
the perpendicular EB, the index of refraction * 
from air to glass being approximately f . 

Ans. 0.098 in. Fig. 37. 

12. A sail ship sails against the wind at an angle of 6o°. The sails 
are set so as to make an angle of 15 with the direction OF of the 

ship. What part of the wind's force is effec- 
tive in producing the forward motion of the 
ship? Ans. 0.183. 

(Suggestion. Let WO be the direction of 
the wind, OS the direction in which the sail 
is set. Decompose a unit of the wind's force 
RO into two components, RP parallel to the 
sail and PO perpendicular to the sail. PR has 
no effect on the sail, and may therefore be disregarded. The other 
component PO may again be resolved into two components, namely, 
PQ perpendicular to the direction of the ship and QO in the direc- 
tion of the ship. PQ is neutral so far as the forward motion of the 
ship is concerned, leaving QO as the only part of the wind's force 
effective in the direction OV.) 

13. A person whose eye is at E, 10 ft. above the level of the water 
PI, observes at / the image of the foot of 
a pile driven in the water. The horizontal 
distance of the observer, from the place 
where the image is formed, is 20 ft., his 
distance from the pile is 65 ft. What is 
the length PF of the pile below the surface 
of the water, the refractive index from air 




Fig- 38. 




to water being approximately |. 



* Index of Refraction = 



Fig. 39- 
Ans. 49.7 ft. 
sine of angle of incidence 
sine of angle of refraction ' 



IOO 



PLANE TRIGONOMETRY 



[chap. V 




14. Two bodies A, weighing 2 lbs., and B, 
weighing 3 lbs., are so placed that B is exactly 
10 ft. west of A. A moves north and B west, 
each at the rate of 12 ft. per second. What 
is the direction and the velocity of their com- "* EB 
mon center of gravity? Fl &- 4°. 

Ans. 33 41'; 8.65 ft. per sec. 
(Suggestion. Locate the center of gravity, C, in two positions, as 
C and C. Find EC and EC, then solve the triangle CEC '.) 

15. The arms of a lever are FA = 2.34 and FB = 5.27 respec- 
tively. At the extremity A of the first arm, a force of 5.34 units 

acts in a direction making an angle of 
a = 63 ° 45 ' with FA produced. What force 
must be applied at B, the extremity of the 
second arm, in a direction making an angle 
(3 = 51 15' with FB produced, in order 
that there may be equilibrium? 




F 




Fig. 41. 



16. Six forces, A = 15, B = 6, C = 5.7, D = 7.9, E = 12.3, 
F = 10, act on the same point and in the same 
plane. 

The angle between A and B is 12 30', 
the angle between A and C is 31 21', 
the angle between A and D is 47 46', 
the angle between A and E is 58 10', 
the angle between A and i 7 is 72 18'. 
Required the resultant force and the direction 
it makes with A. Ans. 50.51, 36 34'. 

(Suggestion. Resolve each force into two components, one along 
PA, the other in a direction perpendicular to PA. Sum the com- 
ponents along each of these directions separately. The sums are the 
rectangular components of the required force.) 




Fig. 42. 



49. Problems in Navigation. In the following problems it is 
assumed that the student is acquainted with the divisions of the 
mariner's compass. On the mariner's compass the total angular 

space about a point is divided equally into 32 divisions, each of 

f\ ° 

which is called a point, that is, a point is equivalent to = n° 15'. 

3 2 



49] 



RIGHT TRIANGLES AND APPLICATIONS 



IOI 



Each point is divided into two half-points, each half-point into two 
quarter-points. In the figure below, the names of the 32 points 
are indicated by their abbreviations. Between north and east the 
points read: 

North by east, north northeast, northeast by north, northeast, 
northeast by east, east northeast, east by north, and similarly for 
each of the other quadrants. 




Fig. 43- 

In the following problems the surface of the earth is considered a 
plane and the distances straight lines, not arcs. By a mile is under- 
stood a sea mile or knot, which is the length of a minute of arc meas- 
ured on the earth's equator so that the earth's circumference measures 
exactly 360 X 60 = 21,600 sea miles. A sea mile is approximately 1^ 
common miles. 

Definitions. The east and west component of a course, or dis- 
tance between two points, is called the departure of the course or 
& distance, the north and south component is called 
the difference in latitude, that is, if WN repre- 
sents any course or distance, and a right triangle 
WSN is formed, by drawing through W a line east 
and west, and through N a line north and south, 
WS is called the departure, and SN the difference 
Fig- 44- in latitude of the course or distance WN. 

In nautical problems difference in latitude is usually expressed in 
degrees and minutes (1 mile = i'). 




102 PLANE TRIGONOMETRY [chap, v 

Exercise 27 

1. A ship sails N.E. by E. at the rate of 8 knots per hour. Find 

the rate at which it is moving due north. 

Ans. 4I knots per hour. 

2. A ship sails S.E. by S. a distance of 578 miles. Find its de- 
parture. Ans. 3 2 1. 1 miles. 

3. A vessel sails W. by S. until the departure is 315 miles. Find 
the actual distance sailed. Ans. 321.2 miles. 

4. A ship sails from latitude 47°3o' N. on a course N.W. by N. 
685 miles. Find the latitude arrived at. 

Ans. Diff. in latitude = 569.6 miles = 569.6' = 9 29. 6'. 
Required latitude = 47 30' + 9 29.6' = 56 59. 6'. 

5. A ship sails S.W. by S. a distance of 1225 miles. Find the 
difference in latitude between the first and last positions of the ship 
and the departure made. 

6. A ship sails from latitude io° 24' N. and after 30 hrs. reaches 
latitude 15 26' N. Its course was N.N.E. Find the average speed 
of the ship. Ans. 6.57 miles per hr. 

7. A ship sails from latitude 35 58' N. on a course between S. and 
E. a distance of 359 miles to a point whose latitude is 32 16' N. 
Find the course of the ship. Ans. S. 51 48' E. 

8. A vessel sails from latitude 5 21' S. on a course N.E. by N. a 
distance of 976 miles. Find the new latitude and the departure. 

Ans. 8° ii'N., 542.3 miles. 

9. A steamer bearing W. by N. with a speed of 12 knots has a 
current setting port * broadside across her track which after 5 hours 
brings her to an island located 108 miles from her starting point. 
Find the true course of the ship. Ans. N.N.W. 

10. One port A is 19 miles due N. of a second port B. Two 
vessels leave the two ports at the same time, one from B sailing 
due E. at the rate of 9 knots an hour, the other from A. The ves- 
sels meet 5 hours out of port. Determine the speed and the course 
of the second vessel. Ans. 9.77 knots, S. 67 07' E. 

11. From a "crow's nest" no ft. above the water, the angle of 
depression of a rock just above the water was found to measure 

* The left-hand side of the ship as one faces ahead, — opposed to starboard. 



So] RIGHT TRIANGLES AND APPLICATIONS 1 03 

15 36'. Find the distance from the rock to the foot of the 
mast. Ans. 394 ft. 

12. A ferry, whose speed in still water is 4 miles per hour, crosses 
a channel whose current is 3J miles per hour. How much will she 
have to- "bear up " in order to make the run straight across, and 
how long will it take her to cross, the channel being 7 miles wide? 

13. An observer on board ship notices that the time between the 
flash of a gun from a fort located N.W. by W. and the report is 5 
seconds. After sailing N.E. by N. the gun was heard again, and this 
time the interval between the flash and the report was 10 seconds. 
Find the distance sailed and the bearing of the fort from the second 
position of the ship. Ans. 9440 ft., S. 63 45' W. 

(Assume the velocity of sound to be 1090 ft. per second.) 

14. A ship sailing due N. observes two lighthouses in a line due 
W., and two hours later the bearings of the lighthouses are found to 
be S. by W. and S.W. by W. respectively. The distance between the 
lighthouses is known to be 10 miles. Find the rate at which the 
ship is moving. 

15. A man-of-war sailing due N.E. at a uniform speed of 20 knots 
observes at 9.30 a.m. a fort bearing N.N.W. Twenty-four minutes 
later the fort is due N.W. Find the distance and bearing of the fort 
from the ship at 10.15 a.m. Ans. 20.54 miles, N. 64 55' W. 

50. Geographical and Astronomical Problems. 

Exercise 28 

1. The shadow of a vertical pole 35 ft. high is 51 ft. long. What is 
the sun's altitude (angle of elevation)? Ans. 34 27.6'. 

2. In Fig. 45, let the circle center E represent the earth and the 
circle center M the moon. The angle PME, 

formed by the line of centers EM and a 
line drawn from M tangent to the earth, 
is known as the moon's equatorial hori- 
zontal parallax and measures 57' 02". EP, 
the earth's mean radius, is 3959 miles. 
Determine EM, the distance of the moon lg ' 45 * 

from the earth. Ans. By use of S and T table, 238,650 miles. 





104 PLANE TRIGONOMETRY [chap, v 

3. In Fig. 45, the angle REM, formed by the line of centers ME 
and a line drawn from E tangent to the moon, is known as the moon's 
angular semidiameter and measures 15' 34". Use the result of the 
last problem and determine RM, the moon's radius. 

Ans. By use of S and T table, 1080.6 miles. 

4. The sun's equatorial horizontal parallax (see Problem 2) is 8.8". 
The radius of the earth is 3959 miles. Find the distance of the 
sun from the earth. Also the sun's diameter, the angular semi- 
diameter being 16' 02". Ans. 92,798,000 miles, 865,620 miles. 

5. The largest angle between Venus and the sun as seen from the 
earth is 47 30'. Using the sun's distance as given in Problem 4, find 
the distance of Venus from the sun, the orbit of Venus being assumed 
circular. 

6. In Fig. 46, EO represents the earth's radius 
( = 3959 miles). Find AP, the radius of the arctic 
circle, latitude 66° 32'. Ans. 1577 miles. 

7. Prove that lengths of two parallels of latitude 
are to each other as the cosines of the latitudes. Fi &- 4 6 - 

8. One degree of longitude on the equator is approximately 69.1 
miles. Determine the length of a degree of longitude at Seattle, 
47 40' N. latitude. Ans. 46.5 miles. 

9. Prove that the lengths of the degrees of longitudes at different 
latitudes are to each other as the cosines of the latitudes. 

10. If one minute of arc of longitude in latitude 6o° measures 
1012.7 yards, how long is the radius of the earth, assuming the earth' 
to be a sphere ? 

11. A ship sails due W. 540 miles in latitude 36 N. What is the 
difference in longitude between the initial and final positions of the 
ship? Ans. g° 40'. 

12. How high above the Pole would an observer have to be to 
have the Arctic Circle for his horizon? (Use the data of Problem 6.) 

Ans. 357 miles. 

13. Beginning at latitude 40 N., two consecutive section lines run 
directly north for a distance of 100 miles. How far apart are they 
at their northern end? Ans. 5166ft. 



So] 



RIGHT TRIANGLES AND APPLICATIONS 



I05 




14. The shortest shadow cast by a vertical rod 25 ft. long at noon 
is 21 ft., the longest shadow cast by the same rod at noon is 56 ft. 

Find the approximate latitude of the place. 

Explanation. The shortest shadow, OS, Fig. 
47, is cast on June 21, the longest shadow, OL, 
on December 21. Halfway between these dates 
the sun will be on the equator; its elevation 
above the horizon at noon will then be the 
latitude of the place, that is, angle OEP in 
the figure, where EP bisects the angle LPS. Ans. 52 59/. 

15. A wall runs east and west; its shadow, measured at right angles 
to the wall, is 10 ft. wide. The altitude of the sun is 25 30', its 
azimuth (angular distance west of the 
south point) is 27 45'. Determine the 
height of the wall. 

(Suggestion. Let WE represent the 
wall, FL the width of the shadow 
measured at right angles to the wall, 
MP the direction of the sun. Then angle 
FMP is the sun's altitude and angle 
MFL its azimuth. The problem in- 
volves two right triangles, namely, MFL and PFM, in which FL, 
angle MFL and angle PMF are given and PF is to be found.) 

16. Each of two observers notices a bright meteor. To the first 
observer, A, the meteor appeared directly south and at an elevation 
of 54 . To the second observer, B, stationed 40 miles west of A, the 
meteor appeared 5 6° east of the South point. On comparing the 
times of observation it was ascertained that the same meteor had 
been observed. Compute the height at which the meteor was seen. 

(Suggestion. In Fig. 48 consider L to be A's position, M, B's 
position, and P the position in which the meteor was observed. 
PF represents the height of the meteor.) 

17. Find the greatest distance at sea at which 
a mountain 14,500 ft. high can be seen, the earth 
oeing considered a sphere, radius 3960 miles, and 
the distance sought being the chord joining the 
point at sea to the foot of the mountain. 

Ans. By S and T functions, 137.4 miles. 





Fig. 49. 



106 PLANE TRIGONOMETRY [chap, v 

(Suggestion. The distance sought is SF = 2 • SO • sin J (SOF) and 

OS 
cos SOF = : , but as OS and OM are nearly equal, it is better to 

OM * 

use the formula in Article 21 for the determination of angle SOF.) 

51. Geometrical Applications. Many geometrical problems can 
be solved by properly dividing the given figures into right tri- 
angles and solving these. Thus, an isosceles triangle is divided into 
two equal right triangles by drawing a perpendicular from the vertex 
to the base; any rhombus is divided into four right triangles by its 
two diagonals; any oblique triangle is equal to the sum or difference 
of two right triangles, formed by drawing the perpendicular from 
any vertex to the opposite side; any regular ploygon is divided into 
as many equal isosceles triangles as the figure has sides, by the lines 
joining the vertices of the polygon to its center; etc. 

Example l Two sides of a rhombus meet at an angle. The 
length of one side is a. Find the lengths of the diagonals. 

Solution. By definition the sides of a 
_^^j^\. rhombus are equal and from geometry it 

A< ^^L j. _^^>(7 is known that the diagonals intersect at 

a""\^|^^""^ right angles and bisect the angles of the 

. D rhombus. 

lg ' ' Let BAD = d be the given angle, and 

AB = AD = a, the given side. 

In the right triangle ABO, two parts, namely, the hypotenuse a 

and the acute angle - are known, hence we may find AO and OB. 

2 

AO = a cos J0, and BO = a sin J 6, 
hence 

AC = 2 AO = 2 a cos \ 6, and BD = 2 BO = 2 a sin \ 6. 
In particular, if the given side is 15 and the given angle 38 , we have 

AC = 30 cos 19 = 28.365, 
BD = 30 sin 19 = 9.768, by the use of natural functions. 

Example 2. The radius of a circle is r. To find the perimeter 
and area of a regular inscribed polygon of n sides. 

Solution. By definition the sides of a regular polygon are equal. 



5i] 



RIGHT TRIANGLES AND APPLICATIONS 



107 



Let O represent the center of the circle and AB one side of the 
inscribed polygon. 

The angular magnitude about O is 360 and, since 

there are n sides, angle AOB = « — . 



n 



Triangle AOB is isosceles, so that if OC is drawn 
from O perpendicular to AB, it will divide the tri- 
angle AOB into two equal right triangles. 

In the right triangle AOC, the hypotenuse equals 




r and the acute angle AOC equals A — = , hence AC and OC 

can be found. 



2 n 



n 



AC = r sin 



180 



OC = r cos 1 



n 



n 



AB = 2 AC = 2 r sin , 



n 



and the perimeter = n ° AB = 2 nr sin 



i8o c 
n 



i8o c 



Also the area of the triangle A OB = J. AB » OC = r 2 sin ±^- cos 



i8o c 



and the area of the entire polygon = n times the area of triangle AOB 

2 . 180 180 
= nr 1 sin cos . 

n n 

In particular, if r — 95 and the figure is a heptagon, n = 7, and we 
have 

Angle ^OC = i^= 25 42 , 5i-5 // . 



Perimeter = 14 • 95 • sin 



i8o c 



A 2 . 180 180 
Area = 7 • 95 2 • sin • cos ■ 



log 14 = 1.14613 

log 95 = 1.97772 

i • 180 , 

log sin = 9-63737-io 

7 

log perimeter = 2.76122 
perimeter = 577.06. 



log 7 = 0.84510 
2 log 95 = 3-95544 

1 • 180° ^ 
log sin = 9.63737 



10 



1 180 

log cos = 9.95471 — 10 

7 " 

log area = 4.39262 

area = 24696. 



108 PLANE TRIGONOMETRY [chap, v 

Example 3. A right pyramid has for its base a square whose 
side is 2 a. The angle formed by a face and the base of the pyramid 
is 6. Determine the altitude, slant height and lateral edge of the 
pyramid, also the angle made by a lateral edge and the plane of the 
base, and the angle between the lateral edge and the edge of the base. 

Solution. Let V-ABCD represent the pyramid, VO its altitude 
and VM its slant height. Join M and 0. Then MA = MO = a, 
and angle VMO = 0. Further, let 

h = altitude VO, 
s = slant height MV, 
I = lateral edge A V, 
a = angle VAO, made by a lateral 

edge and the plane of the 

base, lg ' 52 ' 

jS = angle VAM, made by a lateral edge and an edge of the 

base. 

In the right triangle VMO, a and 6 are given, hence h and 5 may be 
found. Solving, 

h = a tan 6, s = a sec 6. 

In the right triangle VMA, a is given and s has just been found, 
hence I and /3 may be found. Solving, 




]8 = tan -1 - , I = a sec ft or = V# 2 + s 2 . 

a 

Finally, from the right triangle VAO, 

a = sin L - . 

V 

As a numerical example take the side of the base equal to 10, and 
the angle 6 = 6o°. Then a = 5, 

and altitude, h = 5 tan 6o° = 5 v 3, 

slant height, 5 = 5 sec 6o° =10, 
lateral edge, / = V5 2 + io 2 = 5 V5, 

a= sin- 1 ^! = 5o 4 6'o8", 
j8 = tan" 1 2 = 63°26 / o6". 



51] RIGHT TRIANGLES AND APPLICATIONS 109 

Exercise 29 

1. The base of an isosceles triangle is 12 and the angle at the 
vertex is 48 . Find the altitude of the triangle. Ans. 13.48. 

2. The chord of a circle is 20 ft. long and the angle at the center 
subtended by it is 42 io'. Find the radius of the circle. 

Ans. 27.80. 

3. The angle between two lines is 50 21' 24" and a circle whose 
radius is 2380 ft. is tangent to both of them. Find the distance 
from the intersection of the two lines to the point of tangency, — 

(a) When the circle lies in the smaller angle, 

(b) When the circle lies in the larger angle formed by the two 
lines. Ans. 5062.8, 1118.8. 

4. The radius of the inscribed circle of an equilateral triangle is r. 
Find the radius of the circumscribed circle. 

5. A chord of a circle subtends at the center an angle of 8o° 24'. 
In the same circle, how large is the angle subtended by a chord half 
as long? Ans. 37 29.4*. 

(Suggestion. Call the length of the chord a.) 

6. The side of a regular octagon is 7. Find the area. 

Ans. 236.59. 

7. The radius of a circle is r; show that a side of a circumscribed 

o o 

regular polygon of n sides is 2 r tan . 

n 

8. One side of a right triangle is 27.5 and the adjacent acute angle 
is 54 38'. Compute the length of the perpendicular from the ver- 
tex of the right angle to the hypotenuse, and the segments into which 
the hypotenuse is divided. 

9. Solve the preceding problem, using a for the given side and B 
for the given angle. 

Ans. p = a sin B, m = a cos B, n = a sin B tan B, where p is 
the perpendicular, m and n the segments of the hypotenuse, m being 
the segment adjacent to B. 

10. In an oblique triangle two sides and the included angle are 
given, namely a = 25.37, & = 36.12, C = 35 27'. Find the remain- 
ing parts. Ans. A = 43 35.9', B = ioo° 57.1', c = 21.34. 

(Suggestion. Divide the triangle into two right triangles by drawing 
a perpendicular from one of the vertices, A or B, to the opposite side.) 



no 



PLANE TRIGONOMETRY 



[chap. V 



ii. In an oblique triangle one side and two adjacent angles are 
given, namely c = 10, A = 6o°, B = .75°. Find the remaining parts. 

Ans. C = 45°, a = s V6, b = 5 (1 + V 3 ). 

(Suggestion. Divide the triangle into two right B , Q , 

triangles by drawing a perpendicular from B to the 
side opposite.) 

12. A regular parallelopiped has for its base a 
rectangle whose dimensions are AB = 8, A D= 10, 
and its altitude AA' = 15. Find the angles which 5 s 
the diagonal AC makes with AD, with AC and 
with AB. Fl 'S- 53- 

13. A right pyramid has 2 a for an edge of the regular hexagon 
which forms its base and an altitude equal to a. Find the angles 
which a lateral edge makes with an edge of the base, with the plane 
of the base, and the angle which a lateral face makes with the plane 
of the base. 

Ans. tan -1 2 = 63 ° 26', tan -1 J = 26 c 




J4 , tan 



■ ! *v* = 



3 = 30 



14. Verify trigonometrically the following practical rule for in- 
scribing a regular pentagon in a circle: 

Let O be the center of the circle, OA and OC two perpendicular 
radii. Bisect OA in M. Take Mi? equal to MC. With C as center 
and CR as a radius, draw an arc cutting the circle in P. Join C and 
o P. PC will be the side of the pentagon. 

(Suggestion. Assume the radius of length r. 
From triangle OCM find CM = RM. 
From triangle COR find CR = CP. 
From triangle COS (SC=\PC) find angle 
Fig. 54. SOC, which should be 36 .) 




52. Oblique Triangles Solved by Right Triangles. Every 
oblique triangle may be solved by decomposing it into right triangles. 
This is done by drawing a perpendicular from one of the vertices of 
the triangle to the opposite side. In three of the four cases the 
perpendicular can be so chosen that two of the given parts become 
parts of one of the right triangles. This triangle having been solved, 
two parts of the other right triangle become known. The second tri- 
angle may now be solved, and with this all the parts of the original 
triangle become known. The fourth case (given the three sides) 



5*1 



RIGHT TRIANGLES AND APPLICATIONS 



III 



requires a somewhat different method. We will take up each case 
separately. 

Case I. Given one side and two adjacent angles, as b, C, A. 




Fig. 55- 

Let ABC represent the triangle. From A or C, say C, draw a per- 
pendicular CD to the opposite side, or opposite side produced. 

Let CD = p,AD = m, DB = n, angle ACD = 6, angle BCD = <j>. 
Three different figures may arise, — 

the left-hand figure, when A and B are both acute, 
the middle figure, when A is acute and B obtuse, 
the right-hand figure, when A is obtuse. 

In the right triangle A CD, b and angle CAD are known, hence p, m, 
and d may be found. 

From C and 6, <j> may be found. 

Having found <j> and p, we know two parts of the right triangle 
BCD, hence a and n and the angle CBD may be found. 

Knowing m and n, c may be found. 

To check the answers, we repeat the solution, drawing the per- 
pendicular from A instead of from C. 

Case II. Given two sides and the angle opposite one of these sides, 
as a, b, A. 




Fig. 56. 

In this case draw the perpendicular from that vertex which lies 
between the two given sides a and b. 

If a < b, two different triangles exist which have the given parts, 
as in Fig. 56, left, if a = b, only one triangle exists. 



112 



PLANE TRIGONOMETRY 



[chap. V 



In the right triangle CAD, b and A are given, hence p, m and 
(= angle A CD) may be found. 

Next consider the right triangle BCD. p and a are known, hence 
n and <j> (= angle BCD) may be found. 

Finally, 

AB = c = m + n, C=6+(f), B = 180 - (A + C). 

The second solution, if there is one, as in the figure to the left, is 
given by 

AB'=c'=m- n, ACB' =C = 0-<t>, AB'C = B'= 180 - (A+C). 

Check. From triangle A CD, m = b cos A, 

From triangle CDB, n = a cos B, 

and since m + « = c, we must have 

& cos A -\- a cos B = c. 

Case III. Giflew /wo sw/es awd aw included angle, as &, c, ^4. 




pB .4 



^ G— ~ ^ 





Fig. 57- 



Draw the perpendicular from B or C, say C. 

In the right triangle ACD, b and angle CAD are known, hence ^ 
and m may be found. 

Knowing m and c, w may be found. 

Having found p and n, we know two parts of the right triangle 
BCD, hence a and angle CBD may be found. 

Finally, angle ACB = i8o° - {A + angle ,4£C). 

Here, as in Case I, three different figures are possible, according as 

A and 5 are both acute (left-hand figure), 

A acute and B obtuse (middle figure), 

A obtuse (right-hand figure), 
but the above analysis applies to each figure alike. 

A check is obtained by repeating the solution with the perpen- 
dicular drawn from B. 



52] 



RIGHT TRIANGLES AND APPLICATIONS 



113 



Case IV. Given the three sides, a, b, c. 

In the first three cases we were able so to choose the vertex, from 
which the perpendicular was drawn, that one of the right triangles 
contained two of the given parts. In the present case this is not 
possible. The apparent difficulty is easily overcome as follows, — 





Fig. 58. 

In either of the figures 58 we have 

from the triangle CAD, p 2 = b 2 — m 2 , 

from the triangle CDB, p 2 = a 2 — n 2 , 

hence b 2 — m 2 = a 2 — n 2 , 

from which b 2 — a 2 = m 2 — n 2 = (m + n) (m — n). (1) 

We now have, 

in the left figure, in the right figure, 

m-\- n = c, m — n = c, 

and from (1), 

b 2 - a 2 = b 2 - a 2 (b + a)(b-a) 

m — n = = - 1 — — — -j 

m-\- n c c 

, _ b 2 - a 2 _ b 2 - a 2 __ (b + a)(b-a) 

ffl —j— 'yi — — . — — j 

m — n c c 

from which, since a, b, c are given, 

m — n or m-{- n 

may be found. 

We then have in either case, 

(m -f- n) -f- {m — n) 
2 



m, 



(m + n) — (m — n) 

- — ! — - — ' = n. 

2 



We now know two parts in each of the right triangles CAD and 
CDB, and from these the angles A and B may be computed. 



H4 



PLANE TRIGONOMETRY 



[chap. V 



As a check the solution may be repeated with the perpendicular 
drawn from one of the other vertices. 



Example. Given a = 45.652, b = 62.735, c = 51.238; to find the 
angles. 

Solution. 



(b + a)(b- a) 

m — n = - — ■ — — 

c 



b + a= 62.735 + 45.652 = 108.387, 
b- a= 62.735 - 45- 6 5 2 = i7- o8 3> 
c= 51.238. 



log (b + a) = 2.03498 

log(7>- a) = 1.23257 

colog c = 8.29041 



m 



n 



log(w- n) = 1.55796 
m — n = 36.138 * 

{m-\-n)-\- {m — n) __ 51.238 -f~ 36.138 

2 2 

{m-\- n) — {m — n) __ 51.238 — 36.138 



= 43-688, 



From the right triangle CAD, 

— = cos A. 
b 

log m = 1.64036 
colog b = 8.20249 
log cos A = 9.84285 — 10 
,4 =45° 5i' 46", 



7-550- 
2 

From the right triangle CDB, 

- = cos B. 

a 

log n = 0.87795 
colog a = 8.34054 

log cos B = 9.21849 — 10 
B = 8o° 28' 49", 



C=i8o°-(A + B) = 5 3 3 9 f 2 4 ". 

Check. Interchanging b and c in the above formula, we find 

c + a = 51.238 + 45.652 = 96.890, log (c + a) = 1.98628 

c - a = 51.338 - 45.652 = 5.586, log (c - a) = 0.74710 

b = 62.735. colog b = 8.20249 

log (m-n) = 0.93587 
m — n = 8.6272. 

(m + n)-\-(m — n) ___ 62.735 + 8.6272 = - 68l 



m 



n = 



2 2 

(m-\- n) — (m — n) _ 62.735 ~ 8.6272 = 
2 2 



= 27.054. 



* If this result were greater than c, we would have the right-hand figure in 
Fig. 58, and we should have taken m + n = . 



52] RIGHT TRIANGLES AND APPLICATIONS 1 1 5 

— = cos A. — = cos C. 

c a 

. log m = 1.55243 log n = i.43 22 3 

colog c = 8.29041 colog a = 8.34055 

log cos A = 9.84284 log cos C = 9.77278 

^ = 45 5i , 5i ,, ) C = 53° 39' 24", 

B = 180 - (A + C) = 8o° 28' 45".* 

Exercise 30 

Only a few triangles are given here for solution by the method of 
right triangles, for soon we shall study a better method, by means of 
which the computation can in most cases be shortened. 

1. Given a = 342.56, b = 125.72, C = 37 42' 24". 

Ans. A = 124 44' 28", B = 17 33' 08", c =254.97. 

2. Given b = 134.5, c = 235.2, A = 12 f 36.3'. 

Ans. a = 334-7, B = 18 33.9'. 

3. A = 25 25' 25", B = 50 50' 50", c = 278.98. 

yl?w. C = 103 43' 45", a = 123.29, & = 222.70. 

4. C = i27°36.5 / , ^ = 28°3i.3 r , b = 312.9. 

^4w5. c = 612.55, a = 369.22, B = 23 52.2'. 

5. a = 630.50, & = 527.39, .4 = 65°37' 12". 

Ans. B = 49 37' 38", C = 64 45' 10", c = 626.13. 

6. b = 1825, c = 1563, B = 22 13. 7'. Find the remaining parts. 

Arcs. C = 14° 54-8', 4 = 142° 5i-5 r > fl = 2913- 

7. a = 3.537, & = 6.667, c = 5-ooi. Find the remaining parts. 

8. # = 4, & = 5, c = 6. Find the remaining parts. 

<4ws. ^ = 41 24.6', B = 55 46.3', C = 82 49.1'. 

* When checking five-place distances and angles expressed to seconds obtained 
from five-place tables, the results will generally be found to agree only to four 
places for distances and to the nearest tenth of a minute (6") for angles. This is 
because, as we have already observed, the fifth place of a number and the sec- 
onds of an angle obtained from a five-place table are not necessarily accurate. 
When cut down to the nearest tenth of a minute, the results of the two computa- 
tions in the above example agree, each giving 

,1=45° 5i-8', 5 = 8o°28.8 / , C = S3°39A'- 
The solution may therefore be assumed to be correct. 



n6 



PLANE TRIGONOMETRY 



[chap. V 



9. Find the ratio between the sides of a triangle whose angles are, 
■A = 50°, B = 6o°, C = 70 . 

Ans. a :b :c = 0.7660 : 0.8660 : 0.9397. 

10. In a quadrilateral A BCD, Fig. 59, the 
following parts are known : 

,4£ = 673, £C = 589, CD = 223, 
angle B = io5°o6', angle C = 127° 38'. 

It is required to find the length of AD to the i" c 

nearest unit. Ans. 1017. Fig- 59- 




CHAPTER VI 



FUNCTIONS OF AN OBTUSE ANGLE 

Reason for a New Definition. In a right triangle no angle 
can exceed oo°, but when the triangle is oblique one of its angles may 
be obtuse, that is, one of its angles may have any value between 90 
and 180 . In order to solve oblique triangles in the simplest way 
possible, we must define the trigonometric functions for obtuse 
angles. This is best done by means of the conception of rectangular 
coordinates. 

53. Rectangular Coordinates. Let X'X and Y'Y be two lines r 
indefinite in length, intersecting at right angles at 0. The two lines. 

divide their plane into four parts, known 
as the first, second, third and fourth 
quadrants respectively, as indicated by 
the numerals I, II, III, IV, in Fig. 60. 

Let P be any point in the plane of 
the lines X'X and Y'Y, and let PF be 
the perpendicular drawn from P to X'X. 
Join O and P. OP, the distance of P 
from O, is always positive and is designated by r. 

OF, generally represented by x, is called the abscissa of the point 
P. It is considered positive if P is to the right, negative if to the 
left of Y'Y. 

FP, generally represented by y, is called the ordinate of the 
point P. It is considered positive if P is above, negative if below 
X'X. 

Considered together, OF or x and FP or y are known as the rec- 
tangular coordinates of the point P. 

X'X and Y'Y are called the coordinate axes, or axes of reference; 
X'X is the #-axis, or axis of abscissas; Y'Y is the v-axis, or axis of 
ordinates; O is called the origin. 

The abscissa of a point is always written first and the ordinate 
second. Thus, by the point (a, b), we understand the point for 

117 




n8 



PLANE TRIGONOMETRY 



[chap. VI 



which x = a, y = b, that is, the point whose abscissa is a and whose 
ordinate is b. 

From what has been said it is plain that: 

In the first quadrant, x is positive, y is positive, r is positive; 
in the second quadrant, x is negative, y is positive, r is positive; 
in the third quadrant, x is negative, y is negative, r is positive; 
in the fourth quadrant, x is positive, y is y 

negative, r is positive. 
For every point on the #-axis, y = o; 
for every point on the ^-axis, x = o; 
for the origin, x and y are each o. 

Thus, in Fig. 61, 



iY 






Pi 






B 








X' 


A' 






A 








y^> 


B' 






Pn 




T r 


P* 



X 



Fig. 61. 



if for 



Pi, x = 4, ;y = 3; 

x = — 4,^ = 3; then for 5, 

x= - A,y = - 3; for 4', 

* = 4, y = -3; for5 r , 

^ = 4, y = o; for O, 

but each of the distances OP h OPi, OP 3 , OP4 equals V3 2 + 4 2: 
^25 = 5- 



then for P 2 , 
for P 8 , 
for P 4 , 
for ^4, 



^ = - 4, y = °; 

x = o, y= - 3; 
# = o, 3/ = o; 



54. Definition of the Trigonometric Functions of Any Angle 
Less than 180°. 

Let angle XOB = 6 represent any angle less than 180 . Take O 
for an origin, OX for the positive x-axis, and draw OY perpendicular 

to OX. Then OB will be in the first or second 
quadrant according as 6 is acute or obtuse. 
Now take any point P on OB and denote by 
-x r the distance of this point from the origin, 
and by x and y the rectangular coordinates 
of this point with reference to OX and OY 
Fig. 62. as axes> The trigonometric functions of 9 

are then defined as follows, — 




. a y ordinate 

sin 6 = ^ = — , 

r distance 

a x abscissa 
cos 6 = - = — , 

r distance 

a y ordinate 

tan © = ± = -- — : — , 

x abscissa 



esc 6 = 



sec 8 = 



sin 9 



cos 6 



cote 



tanO* 



56] FUNCTIONS OF AN OBTUSE ANGLE 1 19 

It will be observed that when the angle 6 is acute, these definitions 
agree with those given in Art. 7 ; for x, y and r are the base, altitude 
and hypotenuse respectively of the right triangle which we there 
used in defining the trigonometric functions of an acute angle. 

55. The Signs of the Functions of an Obtuse Angle. In the first 
quadrant, x and y, as well as r, are positive. Hence all the functions 
are positive for an angle in the first quadrant. In the second quad- 
rant, x is negative, y and r are positive, hence the ratios - and •*■ are 

r x 

negative, while - remains positive, that is, the cosine and tangent 
r 

and their reciprocals are negative, while the sine and its reciprocal 

are positive. 

The functions of an obtuse angle are all negative, except the sine and 

its reciprocal, which are positive. 

56. Fundamental Relations. Of the six fundamental relations, — 

sin 6 . esc 9=i, sin 2 8 + cos 2 8=1, 

cos 8 • sec 8=1, tan 2 8 + 1 = sec 2 8, 

tan 8 • cot 6 = 1, cot 2 6 + 1 = esc 2 8, 

the first three rest upon the definitions of the cosecant, secant and 
cotangent, and hold, therefore, whether is acute or obtuse, and the 
last three depend upon the relation x 2 + y 2 = r 2 , which is true 
whether x is positive or negative. These six fundamental relations 
hold, therefore, for the functions of obtuse angles as well as for the 
functions of acute angles. 
Also, 

2 

tan 9 — 2- = - = , whether x is positive or negative, hence 

X X COS0 

r 

the two relations, — 

a sin 8 . a cos 8 

tan 8 = - , cot 



cos 8 ' sin 8 ' 

also hold true when 6 is an obtuse angle. 



120 



PLANE TRIGONOMETRY 



[chap. VI 



57. Functions of Supplementary Angles. 

Let XOP = 6 be any angle less than 180 , 

and draw OP' so as to make angle P'OX' p /f 

equal to .angle XOP. Then angle XOP' X '- 

= i8o° - e. 

If OP and OP' are taken equal, the two 
triangles OA P and OA'P' will be geometrically 
equal, and we have 

A'P' AP = 
OP 



sin(i8o°- 6) = 
cos(i8o°- 6) = 
tan(i8o°- 0) = 
csc(i8o°- 6) = 
sec(i8o°- 9) = 
cot(i8o°-6) = 



OP' 

OA' 
OP' 



OA 



OP 



OA 
OP 



A'P' 
OA' 


= 


AP 
-OA 


AP 
OA 




i 




i 


sin (i 


8o° 


-e) 


sin# 




i 




i 


COS (] 


8o° 


-e) 


— COS0 




i 




I 




tan (i8o°— 6) — tan 



= - cot e. 



By comparing our results we observe that the signs ^on the right 
are those of the functions in the second quadrant, hence it appears 
that, — 

Any junction of {i8o° — 6) equals plus or minus the same function 
of 6, the sign being that of the function in the second quadrant. 

The rule just given enables us to express the functions of an 
obtuse angle in terms of the functions of an acute angle, thus: 

sin 1 1 6°= sin (i8o° — 64 ) = sin 64 , 
cos 1 1 6°= cos (180 — 64 ) = — cos 64 , 
tan 1 1 6°= tan (180 — 64 ) = — tan 64 , 
esc n6°= esc (180 — 64 ) = esc 64 , 
sec 1 1 6°= sec (180 — 64 ) = — sec 64 , 
cot n6°= cot (1S0 - 64 ) = - cot64°. 



^V 1 



59] 



FUNCTIONS OF AN OBTUSE ANGLE 



121 




58. Functions of (90° + 8) . Let angle 

p XOP = be any acute angle and draw OP' 

perpendicular to OP. Then angle XOP' = 

±f-x qo° + 0. If OP' is taken equal to OP, the 

triangles A OP and A' OP' are geometrically 

equal, and we have 

A'P' OA ___ e / 



Fig. 64. sin (90 + 8) = 
cos (90 + 8) = 
tan (90 + 6) = 
esc (90 + 6) = 
sec (90 + 8) = 
cot (90 + 8) = 



OP' OP 

OA' - AP 



cos 



OP' 
A'P' 



OP 
OA 



4L 
OP 



= — sin 8, 



= _o^ = _ cote 

OA' -AP AP 



sec 8, 
= — esc 8, 



sin (90 -{- 6) cos 6 



cos (90 + 6) — sin 



= - tan 8. 



tan (90 + 0) - cot B 

Again the signs on the right are the signs of the functions in the 
second quadrant, hence, — 

Any function of (<?o° + 6) is equal to plus or minus the corresponding 
cof unction ofd, the sign being that of the function in the second quadrant. 

This rule, like that of the preceding article, enables us to express 
the functions of an obtuse angle in terms of the functions of an angle 
less than 90 . Thus, 

sin 116 = sin (90 + 26 ) = cos 26 , 
cos 116 = cos (90 + 26 ) = — sin 26 , 
tan 1 1 6° = tan (90 + 26 ) = — cot 26 , etc. 

59. Functions of 180°. If in Fig. 62, 6 is taken equal to 180 , 
OP must coincide with OX', the abscissa of P will be — r and its 
ordinate zero, hence 



o o o 
sin 180 = - = o, 

r 



esc 180 = - = oo, 
o 



cos 180 = 
tan 180 = 



— r 
r 

o 

— r 



= — 1, 



= — o, 



sec 180 = 



= — 1, 



— 1 



cot 180 = 



= — 00. 



— o 



12 2 PLANE TRIGONOMETRY [chap, vi 

The results in the last line need some explanation. The tangent and 
cotangent of every obtuse angle is negative, hence their limiting 
values, as the angle approaches 180 , is negative. Numerically these 
limiting values are o and oo , the minus signs merely indicate that 
these values have been approached through a succession of negative 
magnitudes. 

60. Angles Corresponding to a Given Function. Since the 
cosine, secant, tangent and cotangent of every obtuse angle is nega- 
tive, we can tell whether the angle corresponding to one of these 
functions is obtuse or acute by noting the algebraic sign of the 
function. But the sine and cosecant are positive whether the angle 
is acute or obtuse. In fact, since the sines and cosecants of supple- 
mentary angles are equal in every respect, there will always be two 
angles, one acute and the other obtuse, which will correspond equally 
well to a given sine or cosecant. This is expressed by saying that 
the angle corresponding to a given sine or cosecant is ambiguous. 

Examples. If cos 6 = ^, 6 must equal 6o°, 

and if cos 6 = — J, 6 equals the supplement of 6o°, or 120 . 

If tan 9 = i, 6 must equal 45 , 
and if tan = — i,0 equals the supplement of 45 , or 135 . 
If sin 6 = J,0 may be either 3o°or its supplement, 150 . 

Exercise 31 

1. Locate the points whose coordinates are x = 8, y = 6; x = — 8, 
y = — 6; x = — 8, y = 6;x = 8, y = — 6; and in each case compute 
the distance of the point from the origin. 

2. Locate the points x = $, y = o; x = o, y = $; x = — 1, y = — 1; 
x = o, y = — 1. 

3. The distance of a point from the origin is 2, and its abscissa is 
1 ; find its ordinate and locate the point. 

Ans. Two solutions, y = ± V3. 

4. _The distance of a point from the origin is 10 and its ordinate 
is V 50; find the abscissa and locate the point. 

5. Make out a table containing the sine, cosine and tangent of 
each of the angles 120 , 135 , 150 . 



61] FUNCTIONS OF AN OBTUSE ANGLE 1 23 

6. Construct the angles, having given the following functions: 
cos A = — f, tan B = — 3, sin = \ . 

7. Express the following functions in terms of functions of the 
supplementary angles: sin 115 , tan 165 , cos 125 , cot ioo°, sec 170 , 
sin 145 , cos 136 , tan 95 . 

8. Express in terms of an angle less than 45 ° the following: 

sin 95 , cos 120 , tan ioo°, sec 114 , sin 125 . 

9. Express in two ways in terms of an acute angle, first by means 
of the rule of Art. 57, second by means of the rule of Art. 58, each 
of the following: 

sin 127 35' 13", cos 157 54' 36", tan 140 11' 25.3". 
Which of the two methods is the easier ? 

10. Use the table of natural functions to find the following func- 
tions: sin 112 30', cos 156 25', tan 162 50', sin 105 , cos 175 io', 
tan 126 14'. 

11. being any obtuse angle, prove the following relations: 

sin (0 — 90 ) = — cos 0, esc (0 — 90 ) = — sec 0, 
cos (0 — 90 ) = sin 0, sec (0 — 90 ) = csc0, 
tan (0 — 90 ) — — cot 0, cot (0 — 90 ) = — tan 0. 

61. Review. 

1. (a) What is meant by the logarithm of a number to a given 
base a ? (b) Show that log ab = log a + log b, log a b = b log a. 
(c) What is the logarithm of 1 ? (d) What is meant by a cologarithm ? 
(e) Given log 4 = 0.60206; find log 16, log 2, log J. 

2. (a) What is meant by the common logarithm of a number? 
(b) Give from memory the common logarithms of 10, 100, 0.1, 0.01, 

io r , — , Vio, Vio, Vioo. (c) What is meant by the characteristic 
io" 

of a common logarithm ? What by the mantissa ? (d) Give the 
characteristics of the logarithms of the following numbers: 15, 
153, 6.23, 0.05, 0.0105. ( e ) Give the rule for the characteristic of a 
number greater than 1 ; of a decimal fraction less than 1 . 

3. Explain how a table of common logarithms might be con- 
structed by extracting square roots only. Find the number whose 
logarithm is §, without consulting the table. 






124 PLANE TRIGONOMETRY [chap, vi 

4. Prove that \og a N = °^ 6 ; also show that log&tf • log a b = 1. 

\og b a 

5. What is meant by the principle of proportional parts? Read 
again Art. 32. 

6. Work Problems 13 and 14, Exercise 18. 

7. What is meant by an exponential equation ? Solve Problem 18, 
Exercise 18. 

8. (a) Read again Art. 44. (b) What accuracy is called for in the 
angle of a triangle, when the sides are given correct to three places ? 
To four places? (c) How accurately. can a number be determined 
with the aid of a five-place table of logarithms ? An angle ? (d) When 
should a six-place table be used? When a seven-place table? 

9. (a) How many degrees constitute a point of a mariner's com- 
pass? (b) How many degrees in the angle between N. and N.E. 
by N.? Between N.E. by E. and E.N.E. ? (c) What direction is 
opposite to the direction S.E. by S. ? 

10. Explain how to solve each of the four cases of oblique triangles 
by means of right triangles. 

n. (a) What is meant by the rectangular coordinates of a point? 
(b) The line joining the origin to a point P is 5 units in length and 
makes an angle of 30 with the positive direction of the x-axis. 
What are the rectangular coordinates of the point P ? (c) What are 
the coordinates of the point P, if OP makes an angle of 150 with 
OX? 

12. (a) Define the sine, cosine and tangent of an obtuse angle. 
(b) Prove that sin (180 — A) — sin A, cos (180 — A) — — cos A, 
tan (qo° + A ) = — cot A . (c) Complete the equations 

tan (180 - A) = , sin (oo° + A) = , cos (oo° + A) = . 

13. (a) Find sin 123 , cos 136 , tan io5°3o'. (b) Find x in 
each of the equations: sin x = 0.3423, cos x = — 0.9061, tana? = 
— 0.0913, x being in each case the angle of a plane triangle. 



CHAPTER VII 

PROPERTIES OF TRIANGLES 

In this chapter we shall develop certain properties of triangles 
which will enable us to compute, from a sufficient number of given 
parts, the remaining sides and angles, the area and other magnitudes 
related to the triangle. The principal applications of the results 
developed in this chapter will be treated in a separate chapter. 

62. The Law of Sines. 

(a) First proof. Let ABC be any plane triangle. Draw the per- 
pendicular h from one of the vertices C of the triangle to the oppo- 
site side AB (Fig. 65), or AB produced (Fig. 66). 

c 




Fig. 65. 

In Fig. 65, 
in the right triangle ACD 
h = b sin A, 

and in the right triangle BCD 
h = a sin B. 




In Fig. 66, 
in the right triangle ACD 
h = b sin A , 

and in the right triangle BCD 
h = asm (180 — B) = a sin B. 



Hence, whether the triangle is acute or obtuse, we have 

h = b sin A = a sin B, 



or 



sin A sin B 

Similarly, by drawing a perpendicular from B to the opposite side, 
or the opposite side produced, we obtain 

a _ c 
sin A sin C 

!25 



126 PLANE TRIGONOMETRY [chap.vh 

so that we may write 

a b c , x 

(i) 



sin A sin B sin C 
Equation (i) may be otherwise written thus,— 

a : b : c = sin A : sin B : sin C (2) 

Equation (1) or (2) embodies what is known as the Law of Sines, 
which states that, — 

In any triangle the sides are proportional to the sines of the opposite 
angles. 

(b) Second proof. The Law of Sines may be proven in another 
way, which at the same time brings out the mean- a 

ing of the ratios in equation (1). //\ >s ^' 

Circumscribe a circle about the triangle ABC 
and denote by D the diameter BA ', drawn through 
one of the vertices, as B. Join A' and C. A'BC is ^ 
a right triangle (Why?), and therefore 




a ' At t\ a 

— = sin A ' or D = — — - 

D ' sin A' 



Fig. 67. 



But angle A r = angle A (angles inscribed in the same arc are equal), 
hence 

■^ a a 





sin A' 


sin A ' 


and similarly 








D = \> 

smB 


sinC 


from which 








D- a - 


b c 

= 1 



(3) 

sin A sin B smC 

that is, — 

The ratio of any side of a triangle to the sine of the angle opposite is 
numerically equal to the diameter of the circumscribed circle. 

63. Projection Theorem. 

In Fig. 65, In Fig. 66, 

AD = b cos A, AD = b cos A, 

DB = a cos B. BD = a cos (180 - B)= -a cosB. 
Moreover, 

c = AB = AD + DB. c = AB = AD- BD. 



64] PROPERTIES OF TRIANGLES 127 

Substituting for AD, DB, and BD their values, we have 

c = b cos A + a cos B. c = b cos A — (— a cos B) 

= b cos A -\- a cos 5. 

Hence, whether the triangle is acute or obtuse, we have 

c = a cos B + b cos -4. 

Similarly,* a = b cos C + c cos .B, ■ (4) 

b = c cos A-\- a cos C. 

We may consider the line DB (Fig. 66) the negative of BD, that 
is, DB = — BD. In that case 

c =AB = AD-BD = AD -(- Z>J3)= ^Z> + DB, just as in Fig. 65. 
AD — b cos A is called the projection of ^4 C on AB, 
DB = a cos B is called the projection of CB on AB, 

so that the relations (4) may be stated thus, — 

In any triangle, each side is equal to the algebraic sum of the pro- 
jections of the other two sides upon it. 



64. The Law of Cosines. 




(a) First proof. 




In Fig. 65, 


In Fig. 66, 


b 2 = W + AD 2 


b 2 = h 2 + AD 2 


h 2 = a 2 - DB 2 


h 2 = a 2 - BD 2 


AD 2 = (c - DB) 2 


AD 2 = (c .+ BD) 2 


= c 2 - 2C-DB + DB 2 . 


= c 2 + 2c • BD + .RD 2 . 



Substituting in the first equation the values of h and AD from the 
second and third, we have 

b 2 =a 2 -DB 2 +c 2 -2c • DB+DB 2 b 2 = a 2 -BD 2 +c 2 + 2c - BD+BI? 
= a 2 + c 2 - 2c- DB. = a 2 + c 2 + 2c- BD. 

Now from the figure 

DB = a cos 5. BD = a cos (180 - 5) = - a cos 5. 

1 

* The second formula may be obtained from the first by replacing a by b, b by c, 

c by a, and ^4 by 5, B by C, and C (should it occur) by A. In the same manner 

the third formula may be obtained from the second, and the first from the third. 

That is, if any one of these formulas is given, the other two can be supplied by 

cyclic substitution. 



128 PLANE TRIGONOMETRY [chap, vn 

Substituting these values in the equation just preceding, we obtain 
in either case 

b 2 = c 2 + a 2 — 2 ca cos B. 

Similarly, c 2 = a 2 + b 2 — 2 ab cos C, - (5) 

a 2 = b 2 + c 2 — 2 be cos A. d 

These formulas embody the so-called 

Law of Cosines: In any triangle, the square on any side is equal to 
the sum of the squares on the other two sides diminished by twice the 
product of those two sides times the cosine of the included angle. 

(b) Second proof. The law of cosines may be proved even more 
easily than above by the aid of the projection formulas. We need 
only to multiply the first of the formulas (4) by c, to add a times the 
second and to subtract b times the third. The result is 

c 2 + a 2 — b 2 = c {a cos B + b cos A) + a (b cos C + c cos B) 

— b (c cos A-\- a cos C) 

= 2 ac cos B, 

from which 

b 2 = c 2 + a 2 — 2 ca cos B. 

65. Arithmetic Solution of Triangles. The law of sines and the 
law of cosines are sufficient to solve each of the four cases of 
oblique triangles. 

I. If one side a and two angles are given, the third angle is found 
immediately from the relation A + B + C = 180 . The remaining 
sides may then be found from the law of sines, thus, — 

, flsin^ a sin C 

b = . - , c= ——■ . 

sin A sin A 

II. If two sides b and c and the angle opposite one of them, say B, 
is given, the third side a may be found by the law of cosines; for, 
solving the first of equations (5), considering a as the unknown 
quantity, we find 



a = c cos B ± Vb 2 — c 2 sm 2 B. 

This gives two values for a, as it should, for we know that this case 
has in general two solutions. 



65] PROPERTIES OF TRIANGLES 129 

Having found the third side, the angles A and C may now be 
found from the law of sines, thus : 

0/ • c 

sin A = - sin B, sm C = - sin B. 

b b 

III. If two sides b and c and the included angle A are given, the 
third side a may be found from the law of cosines, for the third 
equation (5) gives 

a = \^b 2 + c 2 — 2 be cos A . 

The sides and one angle being known, the law of sines will give the 
other angles. 

IV. If the three sides are given, the three angles may be found 
from the law of cosines. Thus, to find A, we have from the third 
of the equations (5), 

b 2 + c 2 — a 2 



cos A = 



2 be 



While the law of sines and the law of cosines are theoretically all 
that is necessary to solve triangles, the law of cosines, which would 
have to be used in three out of the four cases, is not adapted to log- 
arithmetic computation. The numerical work necessary to solve 
triangles will be greatly shortened by the use of other formulas 
which we will develop in the following articles. 

Exercise 32 

1. Apply the law of sines to a right triangle and reduce the re- 
sulting equations to their simplest form. 

2. Apply the law of cosines, a 2 = b 2 + c 2 — 2 be cos A, to the cases 
when the angle A = o°, 90 , 180 . 

Ans. A = o°, a 2 = b 2 + c 2 - 2 be = (b - c) 2 . 
A = 9 o°, a 2 = b 2 + c 2 . 
A = 180 , a 2 = b 2 + c 2 + 2 be = (b + c) 2 . 

3. Apply the projection theorem to the case A = 90 ; to the case 
A= B. 

Solve the following problems without the aid of logarithms, — 

4. Given A = 35 , B = 75 , a = 7; find b and c. 

■ Ans. b = 11. 8, c = 11. 5. 



13° 



PLANE TRIGONOMETRY 



[chap. VII 



5. Given A = 65 , b = 10, a = 15; find B and c. 

Ans. B = 37 io', c = 16.2. 

6. Given A = 16 , b = 1$, a = 6; find the remaining parts. 

Ans. B = 43 33.5', C = 120 26.5', c = 18.76, 
or B' = 136 26.5', C = 27 33.5', c' = 10.07. 

7. Given a = 150, & = 200, C = 27 30'; find c. 

^4^5. c = 96.3. 

8. Given a = 2, b = 3, c = 4; find the angles. 

^ws. 4 = 28 57. 3', £ = 46 34.1', c = 104 28.6 r . 

9. By means of the law of sines prove that the bisector of an 
angle of any triangle divides the opposite side into segments pro- 
portional to the adjacent sides. 

10. Derive the law of sines from the law of cosines. 

V7 



(Suggestion. Form the ratio 

V Vi - cosiB 



cosM sin A 
sin.B 



and show that 



it is equal to -•) 
b I 



With a vertex C as a 




66. The Law of Tangents. 

(a) First proof. Let ABC be any triangle, 
center and b, the shorter of the sides 
adjacent to C, as a radius, draw a circle 
cutting BC in P and BC produced in Q. 
Draw AP and AQ. Triangles ACP and 
ACQ are isosceles and QAP is a right 
angle (Why?). Denote the whole angle 
at A by w, and the three parts by x, y, z, 
as indicated in the figure, then angle A PC = x and angle AQC = z 
(Why?). Also 

x-\- y = A, x — y = B, x + z = 90 , x + y + z = w. 

Solving these equations for x, y, z and w, we obtain 

x = i(A + B), y = i(A-B), z = 90 - \ (A + B), 
w = 9 o° + iU-B). 

Now apply the law of sines to each of the triangles APB and AQB. 
thus, — 



AB _ sin (180 — x) __ sin x 
BP sin y sin y 



and 



AB 
BQ 



sin z 

sin ze> 



66] PROPERTIES OF TRIANGLES 131 



or 



c 


_ sin | (A + B) 
sin I ( A - B) ' 

sin | (B + C) 


c 

c* + b 

a 


COS 1 (A + .B) 


a — b 

Similarly, 
a 


COS \{A— B) 
COS 1 (JB + C) 



(6) 



(?)■ 



6 - c sin § (.B - C) 6 + c cos | (jB - C) ' 

& sinj(C+^) & cos|(Q+^) 

c— a sin § (C — -4.) c + a cos|(C — -4.) 

Dividing the first of each pair of equations by the second gives 
a + b tan § ( JL + jg) &+c tan|Cg_+C) 
a - & ~ tan | (^£ -- B) ' b - c ~ tan | (B - C) ' 

c+ « tan|(C + ^l) 

c — a tan I (C — ^i) 

Formulas (7) embody the 

Law of tangents: In any triangle, the sum of two sides is to their 
difference as the tangent of half the sum of the angles opposite is to the 
tangent of half their difference. 

The formulas (6), which we shall have occasion to use hereafter, 
we shall refer to by the name of Double Formulas* 

(b) Second Proof. The law of tangents can be proven more 
easily without the intervention of the double formulas. In Fig. 68 
draw PR parallel to QA, then angle APR = angle QAP = 90 . 
From the similar triangles BQA and BPR, we have 

AQ 
BQ = AQ = AP 
BP RP RP' 

AP 

but BQ = a + b, BP = a - b, 

and ^ = tan APQ = tan x = tan \ (A + B), 

— = tan RAP = tan y = tan \ (A - B), 

* Also called Mollweide's formulas, after the German astronomer (1 774-1825) 
who introduced their use. The cosine form of these formulas appears in New- 
ton's Arithmetica universalis (1707). 



132 PLANE TRIGONOMETRY [chap, vn 

hence 

a+b = tan j (A + B) 
a— b tan § {A — B) 

The law of tangents and double formulas is adapted to the loga- 
rithmic solution of the third case of oblique triangles, that is, when 
two sides and the included angle are given. Suppose the given 
parts are a, b and C. The different steps in the solution are as fol- 
lows: 

1. \ (A -f B) is found from the relation A + B = 180 — C. 

2. J (A — B) is found from the law of tangents, formula (7), 
1st. equation. 

3. Adding and subtracting the results of 1 and 2 we have 

i (A + B) + h {A - B) = A, i(A+B)-i(A-B) = B. 

4. c is found from the double formula (6). 

Having found A and B, c could have been determined from the 

law of sines, thus, — 

a sin C b sin C 

<?= . ,. > or c= , 

sin A sin i> 

but this would require us to look up three new logarithms, namely, 

those of 

a, sin C, sin A , or b, sin C, sin B. 

while the double formula requires but two new logarithms, those of 
sin J (A + B), sin | (^ - B) } or cos J (A + 5), cos J (A - B), 

and these may be taken out at the same time and the same opening 
of the table with the logarithms of tan J (A + B) and tan J (A — B). 

67. Formulas for the Area of a Triangle. 

(a) In terms of the base c and the altitude h. If c represents the 
base of the triangle, Figs. 65, 66, h its altitude and T its area, then 
by elementary geometry 

T = I ch. (1) 

(b) In terms of two sides b and c and the included angle A. From 
the right triangle, Fig. 65 or 66, we have h = b sin A. This value of 
h substituted in (1) gives 

T = I be sin A, - (2) 



6 7 ] 



PROPERTIES OF TRIANGLES 



*33 



(c) In terms of one side c and the angles A, B, C. By the law of 
sines, 



b = 



sin C 



Substituting this value of b in (2) we obtain 

m c 2 sin A sin B 
2 sin C 



(3) 



(d) In terms of the three sides a, b, c. It is shown in plane geometry 
that 



T = Vs (s - a) (s - b) (s - c); 



(4) 



where s = \ (#_+' b + c), that is, s equals half the sum of the three 
sides. 

(e) In terms of s and the radius k of the inscribed circle. 

Let be the center of the inscribed 
circle, Fig. 69. Join O to the vertices of 
the triangle, and draw the radii of the 
inscribed circle to the points of tangency. 
These radii will be perpendicular to the 
respective sides. 

Area of triangle BOC = \ ka, 




s-a F s-b 

Fig. 69. 



Area of triangle CO A 



_ 1 



kb, 



Adding, 



or 



Area of triangle AOB = \ kc. 

T = \ ka + I kb + i kc = \ k (a + b + c), 
T = ks, where s = J (a + b + c). 



(5) 



If the three sides are known separately, (5) enables us to find the 
radius of the inscribed circle of a triangle, for we have 



A 



-!W 



' (s — a) (s — b) (s — c) 



(6) 



on substituting the value of T from (4). 



* This formula is known as Hero's formula for the area of a triangle, after 
Hero of Alexandria (1st century B.C.), who, so far as we know, was the first to 
prove and apply this remarkable formula. 



134 



PLANE TRIGONOMETRY 



[chap. VII 



68. Functions of Half the Angles in Terms of the Sides. 

In Fig. 69, 

AF = AE 
BD = BF 
DC = EC 

Adding, AF + {BD + DC) = BF + {AE + EC), 

and since the sum of the six segments equals a -\- b -\- c = 2 s, 

we have 

AF + {BD + DC) = s, BF+ {AE + EC) = s, 

or AF + a = s, BF + b = s, 

from which 

4F = s - a, BF = s- b. 



(1) 



Also, since the lines AO and 50 bisect the angles A and 5 respectively, 
we have 



, A k 
tan — = — : , 

2 AF' 



. B k 
tan — = — -. 
2 BF 



Substituting for AF and BF their values from (1), we have 



4. A 
tan— = 



a 



tan — = , 

2 s — b 

C Jc 

similarly tan — = ? 

2 s — c 



■ where * = yj (s-a)(s-b)(.s-c) . r (a) 



Again 



^0 2 = OF 2 + AF 2 = k 2 +{s- a) 2 
{s — a){s — b){s — c) -{- s {s — a) 2 _ be {s — a) 
s s 

k 



(3) 



. A OF 

sm ^ == To = Io> 



A AF s-a 
cos — = — = — — 
2 AO AO 



Substituting for AO its value from (3) and reducing the result gives,— 
A = J(j-b){s-c) 



sin 



be 



Similarly sin — = W — 



ca 



A 4 Is {s — a) 

cos — = V/ — 

2 V DC 

B J s{s-b) 

COS— =V/ — -: 

2 * ca 



. C 4 / {s-a){s-b) 
m 7 = V aft ' 



c 4 A (•*— c) 

COS - = V : 

2 ▼ 00 



(4) 



68] PROPERTIES OF TRIANGLES 135 

In each of the radicals the positive root is to be taken, since each 

of the angles — , — , — , is necessarily less than 90 . 
222 

The formulas (2) and (4) are adapted for the logarithmetic com- 
putation of the angles of a triangle when the sides are given. For 
the quantities s, s — a, s — b, s — c, can easily be found, and with 
these known, the right member of each formula involves only multi- 
plications, divisions and the extraction of square roots. 

In general, the angles may be found from either the sine, cosine or 
tangent formulas, but since an angle near oo° cannot be accurately 
found from its sine, nor a very small angle from its cosine, the sine 
formulas are to be avoided when the angle is greater than 45 ° and the 
cosine formulas when the angle is less than 45 °. When all the angles 
are required, the tangent formulas (2) should be used, since they 
require but four logarithms to be taken from the tables, that is, the 
logarithms of s, s — a, s — b and s — c, while the sine and cosine 
formulas (4) require in addition the logarithms of a, b and c. 

Exercise 33 

1. Express in words each of the rules for the area of a triangle in 
Art. 67. 

2. From each of the formulas (2) and (3) in Art. 67, write down 
two others by a cyclic advance of letters. 

3. By comparing the formulas (2) and (4) in Art. 67 show that 



sin A = — Vs (s — a)(s — b)(s — c). 
be 

Why could not the angles be found by this formula as well as by (2) 
or (4), Art. 68? 

4. By comparing the expressions for sin A in Problem 3 with the 

expressions for sin — , cos — in (4), Art. 68, show that 
2 2 

. , . A A 

sin A = 2 sin — cos — . 
2 2 

5. By forming the ratio of sin A : sin B, using the values given in 

Problem 3, show that 

sin A : sin B — a : b. 

This constitutes another proof of the law of sines. 




136 PLANE TRIGONOMETRY [chap, vn 

6. From (3), Art. 62, we have sin A = —, where D is the diameter 

of the circumscribed circle. By substituting this value for sin A in 

(2), Art. 67 show that 

T _ abc 

~ 2D 

7. From (5), Art. 67, we have for the radius of the inscribed circle 

s 
By using the relation (Fig. 70) 

ABC = ABO + ACO - BCO A 

show that 

T T 

tC a/ = , K h = —, 

s — a s — b s — c 

where k a , kb, k c are the radii of the escribed circles, touching the 
sides a, b, c respectively, externally. 

8. Prove that 

L=.L + -L + _L. 

K K a K h K c 

9. Prove that Jc a + 7c b -\- 7c c — Jc = 2D. 

10. Let ABCD (Fig. 71) be an inscribed quadrilateral, a, b, c, d its 
sides and Q its area. Show that 

Q = J (ad + be) sin A . (a) 

c 

Also by comparing the two expressions for the 

diagonal 

h 2 = a 2 + d 2 — 2 ad cos A = b 2 + c 2 — 2 be cos C 

show that 

A = a2 ~ b 2 — c 2 -\- d /, >. 

2 (ad + be) 
and by substituting this value in sin 2 A = 1 — cos 2 A , show that 

sin^ = 2^(s-a)(s-b)(s-e)(s-d) ) (c) 

ad + be 

where s = \ (a + b -f c + d). 

By substituting (c) in (a), show that 




Q = V(s-a)(s- b)(s -e)(s- d). (d) 



68] 



PROPERTIES OF TRIANGLES 



137 



11. It was shown in Art. 65, II, that if two sides b, c, of a triangle 
ABC, and the angle B opposite one of these sides, are given, the 

third side a may be found from the 
relation 




\/b 2 —c 2 sin 2 B 



a = c cos B ± \/b 2 — c 2 sin B. 

Interpret the result geometrically. 

Ans. The two terms on the right 
are respectively the distances from B 



Fig. 72. 

and C to the foot of the perpendiculars drawn from A to BC or BC 
produced. 



CHAPTER VIII 
SOLUTION OF OBLIQUE TRIANGLES 

69. Solution of Oblique Triangles. In the present chapter we 
shall illustrate the computation of each case of oblique triangles by 
a numerical example. Since five-place tables are used in the compu- 
tation, the results are given to only five significant figures and the 
angles to the nearest second. In every case a check has been applied 
to the results obtained. Such a check is to be looked upon as an 
essential step in the solution, since no computation, no matter by 
whom, can be relied upon if it has not been checked. Instead of 
the analytic checks here given, graphic checks are often resorted to 
in practice, but such checks, while more easily applied, are of course 
open to errors of construction and are therefore unsatisfactory, except 
as checks against gross errors. 

The solutions given below are arranged in a form which may serve 
as a model to beginners. It is customary for computers to make out 
a complete schedule of work (as is illustrated in the first case below) 
before referring to the tables, so that when the tables are once 
opened no writing remains to be done except that of filling in the 
numbers taken from the tables. 

70. Case I. Given two angles and one side, as A, B and c. 

(i) To find C, apply the relation A + B + C = 180 . 

(2) To find a and b, apply the law of sines. 

(3) To check, use the double formula. 

Example i. c 

Given y/^^ \a Required 

A = 71 13' 30", /^ \ a= 241.18, 

B = 40 34' 15", a^ — - ^B b = 165.68, 

C= 236.54. Fig ' 73, C=68°I2 , I 5 "r 

Schedule of Work. 

(1) To find C. C = 180 - (A + B) = 68° 12' 15". 

(2) To find a and b. By the law of sines, — 

138 



70] SOLUTION OF OBLIQUE TRIANGLES 139 

a sin A c sin A b sin B , c sin B 

- = or a = — . - = . or = — ; — — ■ • 

c sin C sin C c sin C sin C 

log c = log c = 
log sin A = log sin B = 
colog sin C = colog sin C = 



log a = log b = 

a = b — 

(3) Check.* By the double formula (6), left, 

c sin J {A + 5) , c sin J U - B) 

■ = — H v or a— b = — ; — f-i -t > 

a-b wii(4-B) sin J (A + B) 

iU-B) = 15 19/ 37-, J 04 + B) = 55° 53' 52 /r . 
log c = 

logsini(^-^) = 
colog sin I (^ + -B) = 

log (a — b) = a — b = 

If the computation is correct this value of a — b must agree with 
the value of a — b obtained from (2) above. 

Having completed a schedule as above, we now turn to the tables 
and complete the solution by filling in the missing numbers as 
follows: 

Solution. 

log c = 2.37388 log c = 2.37388 

log sin A = 9.97626 — 10 log sin B = 9.81318 — 10 

colog sinC = 0.03221 colog sinC = 0.03221 

log a = 2.38235 log b = 2.21927 

a = 241.18. b — 165.68. 

Check. 

logc= 2.37388 

log sin \ (A — B) = 9.42214 — 10 
colog sin \ {A -\- B) = 0.08195 

log (a-b) = 1.87797, a-b= 75.50. 

Beginners will do well to follow the above form. Expert com- 
puters save the repetition of recurring numbers such as log c in the 
above example by employing a more compact arrangement. For 
instance, the above solution and check can be put in the following 

* Some authors check by the law of sines, a : b = sin A : sin B, but this 
check is unreliable, for it fails to detect an error in either c or log sin C. 



14-0 PLANE TRIGONOMETRY [chap, viii 



Compact Arrangement 






Solution. 


Check. 


c = 236-S4 


2.37388 


2.37388 


^ = 7 i°i3'3°" 


9.97626 




5 = 40° 34' 15" 


9.81318 




C = 68° 12' 15" 


0.03221 




§ (A - B) = is" 19' 37" 




9.42214 


| (4 + 5) = 55° 53' 52" 




0.08195 


a = 241.18 


2-3&23S 




b = 165.68 


2.21927 




a- b = 75.50 




I.877Q7 



Example 2. Find the area of the triangle given in Example 1. 
Solution. Since one side and two angles are given, we use formula 
(3), Art. 67, 

logc = 2.37388 

log c 2 = 2 log c = 4.74776 

„ . , . „ log sin A = 9.97626 — 10 

T _ c 2 s in A sin £ 1 • 7? 00 

" — 2^m~C — g S = 9- 8l 3 l8 - IO 

colog sinC = 0.03221 

colog 2 = 9.69897 — 10 

log T = 4.26838, T = 18552. 

Exercise 34 

The student must check his results when no answer is given. 

1. Given A = 46 36', B = 54 18', c = 479. 

Find a = 345, b = 396, C = 79° 06'. 

2. Given 4 = 79 59', B = 44 41', a = 79.5. 

Find b = 56.8, c = 66.4, C = 55 20'. 

3. Given ^4 = 54 34', B = 43 56', c = 67.9. Find a, b, C. 

4. Given ^4 = 69 30.2', B = 66° 39. 4', c = 438.3. 

Find a = 592.7, 5 = 581.0, C = 43 50. 4'. 

5. Given ^4 = 29°4i.2', B = 37°5o. 4', a = 32.84. 

Find b = 40.68, c = 61.27, C = 112 28.4'. 

6. Given £ = 78 45.6', C = 63 32. 9', a = 8.875. Find b,c,A. 

7. Given 4 = 64 56' 18", B = 47 29' 11", c = 913.45. 

Find a = 895,14, £ =728.40, C = 67 34' 31". 



71] SOLUTION OF OBLIQUE TRIANGLES 141 

8. Given B =48° 24' 15", C = 31 13' 00", c = 926.74. 

Find 6 = 1337.2, a = 1758.9, A = ioo° 22' 45". 

9. Find the radius of the circumscribed circle in 8. 

Ans. R = 894.06. 

10. Find the area of the triangle in 7. Ans. Area = 301,360. 

11. Find the area and the radius of the circumscribed circle in 2. 
Check by using the relation in Problem 6, Exercise 33. 

71. Case II. Given two sides and the angle opposite one of 
them, as a, 6, A. 

(1) To find B, apply the law of sines. 

(2) To find C, apply the relation A + B + C = 180 . 

(3) To find c, apply the law of sines. 

(4) To check, apply the double formula or the law of tangents. 

When an angle is determined from its sine, as the angle B above, 
it admits of two values which are supplements of each other. 
Whether one or the other or both of these values are to be used 
depends on the conditions imposed by the problem. By construct- 
ing the triangle graphically, it is seen that various cases may arise 
depending on the relations between the given parts a, b and A . 

Construct angle BAC equal to the given angle A, making CA 
equal to b, and from C as a center, with a radius equal to a, draw a 
circle. 

If a is less than the perpendicular distance from C to AB, the 
circle will not cut the line AB. In this case it is impossible to con- 
struct a triangle having the given parts, that is, the given data are 
inconsistent. 

If a is equal to the perpendicular distance from C to AB, the 
circle will be tangent to the line AB at B (Fig. 74), and the resulting 
triangle will have a right angle at B. Now the perpendicular dis- 
tance from C to A B is b sin A , hence in this case a = b sin A . 

If a is greater than the perpendicular distance from C to AB but 
less than b, the circle will cut AB in two points B and B' (Fig. 75), 
and there are two solutions, namely, the triangle ABC and the tri- 
angle AB'C. 

If a is equal or greater than b, the circle will cut the line AB in a 
single point B (Fig. 76), the second point of intersection falling on 
or to the left of A . In this case there will be but one solution. 



142 



PLANE TRIGONOMETRY 



[chap, vin 



So far we have assumed the given angle A to be acute. If A is 
obtuse, a must be greater than b (since A is greater than B, and in 
any triangle the greater angle is opposite the greater side). In this 
case there can be but one solution. (Fig. 77.) 





B 





b sin A < a < b. 

Fig. 75- 



If we disregard the case in which the triangle is right-angled 
(Fig. 74) as not properly constituting a case of oblique triangles, we 
have the following simple test for the number of possible solutions, — 

a = b, one solution, 
a < b, two solutions. 



Example i. 
Given 
a = 34540, 

*= 531-75, 
A = 26°47 , 32 ,/ . 




Fig. 78. 



Required 
£ = 43°56 , oo", 
B'= i36 o 4 / oo // , 
C = 109 16' 28", 
C'= i7°o8'28", 

« = 72345- 
c'= 225.88. 



Solution, a < b, hence there are two solutions. 
(1) To find B and B'. By the law of sines, — 



sin A 
sin-B' 



or sin B = 



b sin A 



log b= 2.72571 
log sin A = 9.65394 — 10 
cologa = 7.46160 — 10 

log sin B = 9.84125 — 10 
B = 43° 56' 00". 



B' = 136 04' 00". 



(2) To find C and C. 
C=i&o°-(A+B) = io 9 ° 16' 28". C =i8o°-(A+B') = ifo$' 28". 



7*1 SOLUTION OF OBLIQUE TRIANGLES 1 43 

(3) To find c and d '. By the law of sines - = , or 

a sin. A 

a sin C , a sin C' 

c= . . ' c = —— — 

sm A sin ^4 

log a = 2.53840 log a = 2.53840 

log sin C = 9.97495 — 10 log sin C" = 9.46942 — 10 

colog sin A = 0.34606 colog sin A = 0.34606 

log c = 2.85941 log d = 2.35388 

c = 723.45. d = 225.88. 

(4) Check. By the double formula, — 

c ^ sm%(B + A) 
b — a sin \ (B — A) 
or 

csinf (B - A) = (b - a) sinf (B + A), 
i(B + A) = 3 5°2i' 4 6", 
i(B-A) = 8° 3 4'i 4 ", 

c'sin \ (B f - A) = (b- a) sin J (B f + A), 
i(B' + A) = 8i 25' 4 6", 

i(B'-A) = 5 4°3Z'i4", 
b — a = 186.29. 

logc = 2.85941 log (b — a) = 2.27019 

log sin J (B — A)= 9.17327 — 10 log sin J (B + A) = 9.76249 — 10 

2.03268 2.03268 

logc' = 2.35388 log (b — a) = 2.27019 

log sin J (B'— A) = 9.91143 — IO lo g sin 2 (&' + A) = 9.99513 - 10 

2.26531 2.26532 

Compact Arrangement 

Solution. * Check. 

b= 531.75 log 2.72571 

A = 26 47' 32" log 9.65394 colog 0.34606 

a = 34546 colog 7.46160 log 2.53850 

B= 43° 5^ 00" log 9.84125 

C = 109 16' 28" log 9.97496 

c = 723-45 lo g 2 -8594i log 2.85941 

i(B-A)= 8° 34' 46" log 9.17327 

i(B + A) = 35 21' 46" colog 0.23751 

b — a= 186.29 c °l°g 7-72981 

0.00000 



144 PLANE TRIGONOMETRY [chap, vm 

W = i 3 6°04 , oo ,/ 

C" = 1 7° 08' 28" log 9.46942 

c' = 225.88 log 2.35388 log 2.35388 

H^-i) = S4W log 9-9H43 

I (B' + A) = 8i°25 , 46 // colog 0.00487 

b — a •■== 186.29 colog 7.72981 

9.99999 
Exercise 35 

Solve the following triangles: 

1. a = 840, 5 = 485, 

Ans. B = 12 14', 

2. a = 41.4, 6 = 52.8, 

Ans. B = 55036' 
5'= 1 24 24', 

3. a = 3.25, 5 = 2.57, 

4. a = 242, 5 = 767, 

Ans. A = io° 55', 

5. a = 91.97, 6 = 93.99, 

Ans. A = 57 23.7', 

6. & = 978.7, c = 871.6, 

Ans. B = 44 01.5', 

£'=135° 58.5 
7.5 = 678.5, c = 423.1, C =53° 23.4'. 

8. a = 48.134, 5 = 35.826, ^4 = 3 6° 24' 00". 

i4w^. 5 = 26 12' 38", C = 117 23' 22", c — 72.022. 

9. 5 = 216.45, c= 177.01, C= 35 36' 20". 

Ans. B = 45 23' 28", A = 99 00' 12", a = 300.29. 
£'=134° 36' 32",^' = 9°47 , o8", a' =51.67. 

10. 5 = 14.332, c= 13.617, C = 45 23' 54". 

11. a = 342.6, 5=745.9, ^ = 43°35.6'. 

^4^5. Impossible. 

72. Case III. Given two sides and the included angle, as 
ce, by C. 

To find A and 5, we first find % (A + B) and \ (A - B). 

(1) To find i (A + B), apply the relation A + 5 + C = 180 . 

(2) To find § (.4 — J3), apply the law of tangents, Art. 66. 



A = 2i° 3 i / . 






C =146° 15', 


c = 


1272. 


4 = 40 19'. 






C = 8 4 ° 05', 


c = 


63.6. 


C'=i5°i7', 


c' = 


16.9. 


^4 = 32° 54'. 






£ = 36° 53'. 






C = I32 12 , 


c = 


947- 


B= 120 35'. 






C= 2 OI.3 , 


c = 


3.85. 


C = 38 14. 2'. 






A = 97°44r3', 


a = 


1395 


.4'= 5° 47- 3', 


a' = 


142. 



72] SOLUTION OF OBLIQUE TRIANGLES 145 

(3) 4 = i (4 + B) +>* (4 - B), B = } (4 .+ 5) - J (4 - J5). 

(4) To find c, apply the law of sines. 

(5) To check, apply one of the double formulas. 

Example i. 

Given Ji Required 

a = 12.346, J/ \a A = 86° 55' 57", 

£ = 5.72i3, ^^ c \ B = 27°33 , 53 ,/ , 

C =65° 30' 10". ^ B c = n.250. 

Fig. 79. 
Solution. 

(1) To find l(A+B). l(A+B) = i (180 - C) = 57 14' 55". 

(2) To find i (A — B). By the law of tangents 

a + b tan § (4 + i?) , 1 / A r,\ a — b . 1 / a , r>\ 

~^ = I — Da n\ ' or tan %U-B)= — — - tan § (4 + 5). 
a — 6 tan ^ (4 — B) a-j- b 

a— b = 6.6247, a-\- b = 18.0673. 

log (a — b) = 0.82117 
colog (a + 6) = 8.74310 — 10 
log tan J (4 + B) = 0.19162 

log tan i(A - B) = 9.75589 - 10 
i(4-5) = 2 9 4i'o2". 

(3) To find 4 and 5. 

4 = i (4 -f B) + J (4 - B) = 86° 55' 57", 

5 =i (4 + 5) - J (4 - 5) = 27 33 ' 53-. 

(4) To find c. From the law of sines we have either 

sin C 7 sin C 
c = a — > or c = — , 

sin 4 sin B 

but since 4 is an angle near 90 it is preferable to find c from the 
second expression. 

log 6 = 0.75749 
log sin C = 9.95903 — 10 
colog sin B = 0.33464 

logc = 1.05116 
c = 11.250. 



146 PLANE TRIGONOMETRY [chap.viii 

(5) Check. By the double formula (Art. 66, (6)), 

c _ sin §(A+B) ^ 
a— b sin \ (A — B) 

logc = 1.05116 log (a — b) = 0.82117 

log sin \ (A — B) = 9.69480 — 10 log sin \ (A + B) = 9.92481 — 10 

0.74596 0.74598 

Compact Arrangement 

Solution. Check. 

a = 12.346 

b = 5.7213 log 0.75749 

a — b = 6.6247 logo.82117 logo.82117 

a + b = 18.0673 colog 8.74310 

C = 65 30' 10" log sin 9.95903 

I (A + B) = 57 14' 55" log tan 0.19162 log sin 9.92481 

\ (A — B) = 29 41' 02" log tan 9.75589 colog sin 0.30520 

A = 86° 55' 57" 

■B = 27 33' 53" colog sin 0.33464 

c = 11.250 1.05116 1.05118 

Exercise 36 

1. Given a = 486, b = 347, C = 51 36'. 

Find A = Z3 15', B = 45° 09', c = 383.5- 

2. Given a = 364, 6 = 640, C = 53 14'. 

Find A = 34° 38', ^ = 92° 08', c = 513. 

3. Given a = 875, 5 = 567, C = 34 52'. Find A, B, c. 

4. Given a = 233.4, 6 = 557.2, C - 18 23.0'. 

Find A = 12 22. o', i? = 149 15.0', c = 343.7. 

5. Given b = 145.9, c = 39.90, A = 92 11.3'. 

Find B = 72 40.7', C = 15 08.0', a = 152.7. 

6. Given c = 453.9, a = 478.1, B = 35° 37-9'- Fi nd C, 4, 6. 

7. Given a = 51.269, b = 14.687, C = 62 og' 24". 

Find A = 101 32' 32", B = 16 18' 04", c = 46.269. 

8. Given b = 467.92, c = 612.34, ^4 = 45 29' 16". 

Find 5 = 49 34' 05", c = 84 56' 39", a = 438.36. 



73] 



SOLUTION OF OBLIQUE TRIANGLES 



147 



9. Given c = 345- 6 7, a = ^>54-3 2 , B = 67° 45' 45"- Fin d C, A, b. 
o. Given a = 447.45, b =216.45, C =116° 30' 20". Find the area. 

2fw. r = 43336. 

11. Show that when the included angle is a right angle, the law of 
tangents gives 

a — b 



tan \ (A - B) = 



a + b 



73. Case IV. Given three sides, a, 6, c. 

Each of the angles A, B and C is found by applying one of the 
formulas (2) or (4), Art. 68, for the tangent, sine or cosine of half the 
respective angle, but for the reasons stated in Art. 68 the tangent 
formulas are generally to be preferred. 

To check, apply the relation A + B + C = 180 . 

Example i. 




Given 

a= 12.653, 

b = 17-213, 
c = 23.106. 

Fig. 80, 

Solution. By formula (2), Art. 68, 

k ,___ B 

2 



Required 

A = 32° 3 6 / 22 / ' 
B = 47 08' 42" 
C = ioo° 14' 56 



tan — = 

2 s — a 



tan — = 



5- b 



tan — = 



s — c 



where 



s = 



a + b + c 



s = 26.486 
s - a " - 13.833 
s- 6= 9.273 

s — c = 3-3 8 o 



(5 — a) (s — b) (s — c) 
s 

cologs = 8.57698 — 10 

log (s — a) = 1. 14092 

log (s — b) = 0.96722 

log (s — c) = 0.52892 

log& 2 = 1. 2 1404 

log k = 0.60702 



log £ = 0.60702 

log (s— a) = 1.14092 

log tan \A = 9.46610 

J,4=i6 i8' n ; 

^= 3 2°36 / 22 > 



log £ = 0.60702 

log (s—b) = 0.9672 2 

log tan \B = 9.63980 

i£=23° 3 4 , 2i' 
5 = 47° 08' 42' 



log& =0.60702 

log (s—c) = 0.52892 

log tan \ C = 0.07810 

iC=5o°o7 / 28" 

C=ioo°i4 / 56". 



Check. A + B + C=32° 36' 22 ,, + 4 7° 08' 42"+ ioo° 14' 56" = 180 . 



148 



PLANE TRIGONOMETRY 



[chap. VIII 







Compact Arrangement 










Solution. 




Check. 


a= 12.653 










b = 17.213 










c = 23.106 










s = 26.486 




colog 8.57698 






s-a= 13.833 




log 1. 14092 






s-b = 9.273 




log 0.96722 






s — c= 3.380 




log 0.52892 






k 2 




log 1. 2 1404 






k 




log 0.60702 






\A = i6°i8' 


11" 


log tan 9.46610 


A = 


32° 3 6' 22" 


iB=2 3 ° 34 f 


21" 


log tan 9.63980 


B = 


47 08' 42" 


iC = 5o°o 7 ' 


28" 


log tan 0.07810 


C = 


ioo° 14' 56" 
180 00' 00" 



Example 2. Find the area of the triangle and the radii of the 
inscribed and escribed circles for the triangle in Example 1. 

Solution. a = 12.653, b = 17.213, c = 23.106. 

The area of a triangle in terms of the sides is given by (4), Art. 67. 

T=V s (s- a) 0- b) (s- c). 

The formulas for the radii of the inscribed and escribed circles are 
given in Problem 7, Exercise 33. 

_ _T 7 T 

S 



k = ?, 



kh = 



s — a 



s- b 
Using the results of Example 1, we have 

logs = 1.42302 
log (s — a) = 1. 14092 
log (s — b) = 0.96722 
log (s- c) = 0.52892 



Rf. 



s — c 



log T 2 = 4.06008 
log T = 2.03004 
log k = 0.60702 

k>g& a = O.88912 

log&b = 1.06282 

log& c = 1. 501 1 2 



T = 107.16, 
k = 4.046, 
ka = 7-747; 
h= n.55 6 > 
k c = 31.704. 




Fig. 81. 



73] SOLUTION OF OBLIQUE TRIANGLES 1 49 

Check. A convenient check is obtained by using the relation in 
Problem 8, Exercise 33, 

R, Kd K\) K c 

log - = colog k = 9.39298 — 10 - = 0.24716 

k k 

log— = colog k a = 9. 1 1088 —10 — = 0.12909 

Ra %a 

log — = colog k b = 8.93718 - 10 j- = 0.08653 

Kb K b 

log— = C0l0g^ c = 8.49888 — IO — = O.O3154 

1 1 = 0.24716 =-• 

K a K}j K c H 



Exercise 37 

.1. Given a = 286, b = 321, c = 463. 

Find ,4 = 37 34', 5 = 43° "',C = 99° if> 

2. Given a = 3.21, b = 3.61, c = 4.02. 

Find A = 49 24', B = 58 $&', C = 71 58'. 

3. Given a = 74.6, b = 81.9, c = 90.0. Find the angles. 

4. Given a = 354.4, b = 277.9, c = 401.3. 

Find ,4 = 59°39-5'>£ = 42° 35-3', C = 77° 45-*'- 

5. Given a = 1.961, b = 2.641, c = 1.354. 

Find ,4 = 46 o$.f , B = io 4 o7.6 / , C = 29° 4 8.8'. 

6. Given a = 87.06, b = 9.16, c = 79.02. Find A, B, C. 

7. Given a = 3359.4, b = 4216.3, c = 4098.7. 

Find A = 47 38' 00", B = 68° oi' 06", C = 64 20' 54". 

8. Given a = 33.112, b = 44.224, c = 55.336. 

Find A = 36 45 r 14", B = 53° 03' 08", C = 90 n' 3 8 ,r . 

9. Given a = 14.493, & = 55-436, c = 66.913. Find the angles. 

10. Given a = 46.78, 6 = 35.90, c = 77.00. Find the area. 

Ans. T = 573.91. 
it. Find the area in Problem 2. Check. 



150 PLANE TRIGONOMETRY [chap.vih 

12. Find the radii of the inscribed circle, of the escribed circles 
and the circumcircle of the triangle in problem 10. 

Check by using the relation k a + kb~\- k c — k = 2 D = 4 R, Prob- 
lem 9, Exercise 33. 

13. a = 4, b = 5, c = 6. Find the angles by applying the law of 
cosines directly, that is, 

\P" — I— C^ — Q? 

cos A = — — - , cos B = etc. 

2 be 

14. The sides of an inscribed quadrilateral taken in order are 

= 5M = S3, c = l6 > d = 6 3- 
Required the area Q of the quadrilateral and the angles A, B, C, D, 
the notation being that employed in Fig. 71, Problem 10, Exer- 
cise 7,7,. 

Ans. Q = 1428, A = 44° 46', B = 90 , C = 135 14', D = oo°. 

74. Practical Applications. The solution of oblique triangles 
finds numerous applications in the various arts and sciences. Chief 
among these are surveying, engineering, physics, astronomy and 
navigation. The simpler applications which involve the solution of 
a single triangle need no explanation, since they may be immediately 
referred to some one of the four cases treated in the preceding sec- 
tions. Many of the practical applications, however, involve several 
triangles which must be successively solved in whole or in part 
before the required distance or angle can be ascertained. Some- 
times the intermediate triangles to be solved are not apparent from 
the figure, but must be sought by some auxiliary geometrical con- 
struction. Again, it may happen that no single triangle exists con- 
taining the requisite number of parts; in such cases the solution is 
effected by solving the equations which arise by applying the for- 
mulas of Chapter VII so as to involve the unknown parts. We shall 
illustrate each of these cases by an example. 

(a) System of triangles. Example i. Given one side c of a 
quadilateral ABC'C (Fig. 82) and two angles A and B adjacent to 
this side, also the angles a, /3 which the diagonals d h d 2 , drawn from A 
and B respectively, make with the given side, to determine the side 
x opposite the given side c. This problem is sometimes known as 
Hansen's problem. 



74] 



SOLUTION OF OBLIQUE TRIANGLES 



J5 1 




Analysis. Let AC = a, BC = b, 
angle CAC = a, angle CBC = 0', 
angle ACC = 6, angle BCC = d' , 
angle ACC = <f>, angle BCC = </>'. 

(i) In the triangle ABC, one side c and 
two adjacent angles A, (3 are known, hence 
J 2 may be found. 

(2) In the triangle ABC', one side c and two adjacent angles a, B 
are known, hence b may be found. 

(3) Then, in the triangle CBC', two sides d 2 , b and the included 
angle /3' = B — /3 * are known, hence # may be found. 

(4) Check. Compute x again, using the triangle CAC', having 
previously found a and d\ from the triangles ABC and ABC re- 
spectively. 

Illustration. In order to determine the length CC (Fig. 82) of a 
trestle to be built across the end of a lagoon, a distance AB, 500 ft. 
long, was measured off, and the following angles were measured 
with a transit: 

CAB = 105 30' = A, CBA = 95 50' = B, 
C'AB= 35°i7 , = « 3 CBA = 47° 32' = (3. 

Required the distance CC = x. 
Solution. (1) Triangle ABC. By the law of sines 

sin A ! sin 



do = c 



and a 



sin ACB' sin ACB 

ACB = 180 - (A + fi) = 26 58'. 
logc = 2.69897 = 2.69897 



log sin A = 9.98391! 
colog sin^4CI> = 0.34345 

log ^ 2 = 3-02633 
di = 1062.5. 



log sin/3 = 9.86786 

= Q-34345 

log a = 2.91028 
a = 813.36. 



* It is assumed that the points A, B, C, C are in the same pfane, otherwise 
the angles CAC and CBC must be given in addition to the data of the problem. 

f 9.98391 — 10. In this and the following problems the — io's are omitted 
when this may be done without danger of confusion. This is a common practice 
among computers. 



152 PLANE TRIGONOMETRY [chap, vm 

(2) Triangle ABC' . By the law of sines 

7 sin a j j sin B 

b = c — — , and di = c 



sin AC B' sin AC B 

AC'B = 180 - (B + a ) = 48 53'. 

logc = 2.69897 = 2.69897 

log sin a = 9.76164 log sin B = 9.99775 

colog sin AC'B = 0.12299 =0.12299 



log 6 = 2.58360 log^i = 2.81971 

* = 383-35- di = 660.25. 

(3) Triangle CBC . Triangle CAC. 
By the law of tangents 

tan i^z£ = <k=b tan H+£ i tan ^J L = a I1 i i tan «+_« 

2 d 2 + 6 2 2 a + ^i 2 

I (#'+*0 = i (i8o°-« = 4 8° 18'. i (<£+0) = f (i8o -a / ) = 54 53-5 / . 

d 2 — 6 = 679.15, d 2 +6= 1445.85, a— ^i = i53- II J <H-^i=i473- 6l > 

log (d 2 — ft) = 2.83106 log (a — 6^1) = 2.18501 

colog (^2 + b) = 6.83988 colog (a + <Zi) = 6.83162 

log tan J (<£' + 0') = 0.34836 log tan J (0 + (9) = 0.15302 



log tan J (<j)' — 6') = 0.02020 log tan J ((f) — 6) = 9.16965 

\ W - e 1 ) = 46 19' 55"- 1 (4> - 9) = 8° 24' 25". 

By the double formula involving the sines, 

x=((h - b) ^iM±n x=(ffl _, l)S mig+|l. 

sm f (9 — a ) sin f (9 — 0) 

log (^2 — 6) = 2.83196 log (a — Ji) = 2.18501 

log sin J (cf) f -f- 0') = 9.96022 log sin J ((f) -f- 0) = 9.91278 

colog sin J (<// — 6 f ) = 0.14065 colog sin| (<f> — 6) = 0.83505 

logx = 2.93283 logx = 2.93284 

x = 856.7, x = 856.7 (check). 

(b) Auxiliary geometrical constructions. Example 2. In Example 1 
it was shown how the distance CC may be found without leaving 
the line AB. Another important problem is to determine one's 
position from the angles which the sides of a known triangle subtend 
from that position. This is known among surveyors as the three- 




74J SOLUTION OF OBLIQUE TRIANGLES 1 53 

point problem, sometimes as Pothenot's problem* It may be stated 
thus: Given three points A, B, C whose mutual distances are known, 
to find the distance of a fourth point P from either 
of the points A, B, C, having given the angles, which 
the sides AB, BC, CA subtend at P. 

Analysis. Let A, B, C be the three points whose 
mutual distances are, — 

AB = c,BC = a, CA = b, 

and let P represent the fourth point at which AC and Fl S- 8 3- 
BC subtend the angles a and (3 respectively. It is required to find 
the distances d 1} a\, d 3 of P from A , B and C respectively. 

Circumscribe a circle about the triangle ABP, and let C be the 
point at which this circle cuts PC or PC produced. Draw AC and 
BC ' . Then angle BAC = angle BPC = (3, being inscribed angles 
subtended by the same arc, and likewise angle ABC' = angle A PC 
= a. 

(1) In the triangle ABC', one side c and the two adjacent angles 
a, /3 are known, hence AC can be found. 

(2) In the triangle ABC, the sides are known, hence A, the angle 
opposite the side a, can be found. 

(3) In the triangle ACC ', two sides b and AC and the included 
angle CAC = A — (3 are known, hence angle ACC = 71 can be 
found. 

(4) In the triangle A PC, one side b and two angles a, 71 are known, 
hence the sides d\, d 3 can be found. 

(5) Similarly, d 2 and d 3 can be found from the triangle BPC, 
having previously computed the triangle BCC . 

(6) Check. Compare the value of d 3 found in (4) with the value 
ofd 3 in(5). 

Illustration. A, B, C (Fig. 8$) are three hostile forts whose 

mutual distances are known to be AB = 4 miles, B'C = 2 miles, 

CA = 3 miles. From a battery planted at P, AC subtends an angle 

34 30' and BC an angle of 23 45'. Find the distance of the battery 

from each of the forts. 

* Pothenot's problem and Hansen's problem are named after the men who 
were supposed to have first formulated and solved these problems. It is now 
known that both of these problems were previously solved by Snellius, the first 
in 161 7, the other in 1627. Hence, if any one's name is to be associated with 
these problems hereafter it ought to be that of Snellius. 



154 PLANE TRIGONOMETRY [chap, vm 

Solution. With the notation indicated in the figure, we have 
given 

a =2, b = 2>, c = 4, a = 34° 30', = 23 45'; 

to find di f (I2, ds. 

(1) Triangle ABC. By the law of sines 

\ ^, sin a sin a t.^, sin 8 

AC —c — = c- ; -> BL =c 

sin AC B sn(a + /3) sin {a + 0) 

a + = 34 o 3o , + 23 o 45 , = 5 8 o i5 / . 
log c = 0.60206 = 0.60206 

log sin a; = 9.75313 log sin = 9.60503 

colog sin (a + 0) = 0.07040 = 0.07040 

log^C" = 0.42559 log£C" = 0.27749 

4C" = 2.6644. BC = 1.8945. 

(2) Triangle ABC. By the law of cosines 

. b 2 + c 2 — a 2 „ c 2 + a 2 — b 2 
cos A — — cos B = — ! 

2 be 2 ca 

= *' + 4 2 - ^ = .8 750 _ 4 2 + * - 3 2 = a6g 

2 • 3 • 4 2 • 4 • 2 

4 - 28 57'. . ' 5 = 4 6° 34'. 

(3) Triangle ^CC'. Triangle BCC. 

By the law of tangents 

2 2 

= ^ig tan ^dbS, = ^J^ tan ^C + 7 2 . 

6 + i4C 2 a + BC 2 

CAC = A- (3=5° 12', CBC= B - a= i2°o 4 / . 

^C ; C+7« _ i8o°-5° "' -oy ,,/ ggg±7g _ i8o -i2°04 / _ 0?0 fP/ 
-—57 24, — 53 55, 

22 22 

6 - 4C' = 0.3356, a - £C = 0.1055, 

b + AC= 5.6644. a + BC = 3.8945. 

log (6 - AC) = 9-52582 log (a - 5C) = 9-02325 

colog (b + i4C) = 9.24685 colog (a + J5C') = 9.40954 

, . ACC + 71 o H 1 , 5CC + 72 

log tan — = 1.34285 log tan ! — - = 0.97596 

2 2 



74] 



SOLUTION OF OBLIQUE TRIANGLES 



155 



, . AC'C - 71 

log tan — = 0.11552. 

2 



log tan ^ = 9.40875 

2 

BCC ~ 72 = 14° 23' 



„ _ AC'C + 7l 
2 

= 34° 5 2 ' 



AC'C - 71 



(4) Triangle ^PC. 
By the law of sines 



sin a 



^ _. £C'C + 72 

72 — ■ 

2 

= 69° 35'- 

Triangle SPC. 

smjS 



^C ; C - 72 



d = b sinCAP = b sin ( a + 7i) = fl sin Qg + 72) # 
sin a sin a sin (3 

a + 7i = 34° 3°' + 34° 5 2 ' = 6 9° 22 ', 
/3 + 72 = 23*45' + 69° 35' = 93° 2 o'- 



log& = 0.47712 

log sin 7! = 9.75714 

colog sin a = 0.24687 

logJi = 0.481 13 
di = 3.0277. 

logft = 0.47712 

logsin(a + 7i) = 9.971 21 

colog sin a = 0.24687 

log^ 3 = 0.69520 

^3 = 4-957- 



log a = 0.30103 

log sin 72 = 9.97182 

colog sin/3 = 0.39497 

log d 2 = 0.66782 

^2 = 4-6539- 

log a = 0.30103 

log sin (0 + 72) = 9-999 2 5 

colog sin/3 = 0.39497 

log d s = 0.69526 



dz = 4.957 (check). 



(c) Solution by solving a system of simultaneous equations. 

Example 3. From a point O (Fig. 
84), the segments AP, PQ, QB of a 
straight line subtend the angles a, 7, 
/3 respectively. The distances AB = d, 
and PQ = c are known; required the a[ 
distances AP = x and QB = y. 

Solution. In this case no single tri- 
angle contains a sufficient number of known parts to afford a 
solution, but on applying the law of sines to the various triangles, 
we obtain the following equations, — 




Fig. 84. 



156 PLANE TRIGONOMETRY [chap, vm 

x OP f v x + c 00 , x 

sin a sin ^. sm (a + 7) sin ^4 

^_=^2-, (3 ) y+ c = -2g.. (4 ) 

sin /3 sin 23 sin (/3 + 7) sin 5 

Dividing (1) by (2), and (3) by (4), we obtain 

x . sin (a + 7) = QP^ , ^ y # sin (g + 7) = Og # ^ 

x + c sin a 0<2 y + c sin /3 OP 



Multiplying (5) by (6) gives, after a slight reduction, 

ary sin a sin ft 

(x + c)(y + c) sin (a + 7) sin (/3 + 7) 



(7) 



From the conditions of the problem we also have 

x + y = d — c. (8) 

The simultaneous equations (7) and (8) contain but two unknowns, 
x and y, which may be found from them by the familiar methods of 
algebra. 

Second solution. Call the common altitude of the triangles h 
(Fig. 84). The area of each triangle may be expressed in two ways, 
first as one-half the product of the base by the altitude, second as 
one-half the product of two sides into the sine of the angle included 
by these sides (Art. 67, (2)). 

Accordingly we have 

2 • area APO = xh = AO • PO • sin a (9) 

2 • area QPO = yh = QO • BO • sin (10) 

2 • area PQO = ch = PO ■ QO • sin 7 (11) 

2 • area ABO = dh = AO • BO • sin (a + 7 + /?) (12) 

Dividing the product of (9) and (10) by the product of (n) and 
(12), and canceling the factors which appear in both the numerator 
and the denominator, gives 

xy _ sin a sin ft _ , sin a sin (3 , n 

cd sin 7 sin (a + j8 +-7) sin 7 sm (a + j(3 + 7) 

Equation (13), together with equation (8), is sufficient for the 
determination of x and y. 

Illustration. An island PQ (Fig. 84) one mile wide lies in a direct 
line between two cities A and B on opposite shores of a river. From 



74] SOLUTION OF OBLIQUE TRIANGLES 1 57 

a point up the river the channels AP and QB subtend angles of 
45° 35' and25° io' respectively, while the island subtends an angle of 
8° 28'. The cities are known to be 1 1 miles apart. Find the widths 
of the channels. 

Solution. With the notation in Fig. 84, we have 

d = 11, c = i, a = 45° 35', P = 25° 10', 7 = 8° 28'. 

a + y= 5 4°o 3 ', (3 + 7 = 33° 3^. 

log sin a = 9.85386 

log sin/3 = 9.62865 

colog sin (a + 7) = 0.09177 

colog sin (/3 + 7) = 0.25659 

, sin a sin 6 , xy 

log ■ , . \ . fQ , v = 9-83087 = log , - rf - : 
sm (a + 7) sin (jS + 7) (% + c)(y + c) 



xy 



= 0.67743 = k, x + v = d — c = 10. 



(> + c)(;y + c) 
EHminating v between these two equations gives 

9 , 11 k 

x 2 — ioxH = o. 

1 — k 

Putting for k its value, we have the equation 

x 2 — iox+ 23.10 = o, 
from which x = 3.62, or 6.38, 

and hence y = 6.38, or 3.62. 

Check. We compute xy by the second method and compare this 
product with the value of xy from the results just obtained. 

xy = 3.62 X 6.38 = 23.10. 
Applying (13), 

a + + 7 = 45° 35' + 25° 10' + 8° 28' = 79 13'. 

log c = 0.00000 

\ogd = 1.04139 

log sin a: = 9.85386 

log sin/3 = 9.62865 

colog sin 7 = 0.83199 

colog sin (a + |S + 7) — 0.00774 

logry = 1.36363, xy = 23.1O0 



158 PLANE TRIGONOMETRY [chap, vin 

In the following exercises the student is expected to check his 
results whenever no answer is given. Reread Article 44 for the 
number of significant figures to be retained in the answers. Only as 
many problems need be worked out in detail as are necessary to 
make the student reasonably familiar with the methods of com- 
putation with logarithms; after that an additional number of prob- 
lems may be selected to be analyzed only. The student is not 
expected to work or analyze problems which involve principles from 
physics or astronomy with which he is not familiar. 

75. Miscellaneous Heights and Distances. 

Exercise 38 

1. A man walking along a straight road, observes a church at an 
angle of 45 with the direction of the road. After walking another 
mile, the same church makes an angle of 30 with the opposite direc- 
tion of the road. How far is the church from the road? 

Ans. 0.37 miles. 

2. The angle of elevation of an aeroplane from a point due west 
of it is 50 , and from a point due east of it is 43 °. The distance 
between the two points is 1000 ft. Find the approximate height 
of the plane. Ans. 523.2 ft. 

3. Two mountains are 9 and 13 miles respectively from a town, 
and the angle subtended by them is 7i°4o\ Find the distance 
between the mountains. 

4. From a point 3 miles from one end of an island and 7 miles 
from the other end, the island subtends an angle of 3$° 45'- Find 
the length of the island. Ans. 4.8 miles. 

5. From a town two roads making an angle of 54 cross the same 
river, the first at a distance of 8 miles," the other at a distance of 
12 miles. Find the distance between the points where the roads meet 
the river. Ans. 9.75 miles. 

6. The distances between three ports A, B, C, are as follows: 
AB = 71.2 miles, BC = 28.9 miles, CA = 60.1 miles. C is due 
north of A ; in what direction from A is B ? Ans. N. 23 ° 32' E. 

7. Two stations A and B on opposite sides of a mountain are 
both visible from a third station C. The distances AC, CB and the 
angle ACB were measured and were found to be n. 5 miles, 13.4 
miles and 59 34' respectively. Find the distance from A to B. 



75 J SOLUTION OF OBLIQUE TRIANGLES 1 59 

8. The two slopes of an embankment measure 48.5 ft. and 84.0 ft. 
respectively. The inclination of the first slope is 21 31'. Find the 
inclination of the second slope and the width of the embankment at 
its base. Ans. 12 14', 127 ft. 

9. What is the largest circular track that can be laid out within 
a triangular field whose sides are 229 yards, 109 yards and 312 yards 
respectively? Ans. Radius of track = 28.8 yards. 

10. A person walking along a straight road observes the angle of 
elevation of the summit of the hill to be 25 . After walking 2 miles 
on an incline of io° towards the summit, the angle of elevation is 
observed to be 37 . Find the air-line distance of the summit from 
the second position of the observer. 

11. A person standing on the bank of a river observes the angle 
subtended by a tower on the opposite bank of the river to be 65 oo', 
and when he recedes 50 ft. from the river, he finds the angle to 
be 3 2 oo'. Find the height of the tower and the width of the 
river. 

Suggestion. Call the height of the tower x, and the width of the 
river y. Set up two equations involving x and y and solve. 

Ans. x = 44.1 ft. y = 20.6 ft. 

12. From a point A two straight roads extend at an angle of 28 30'. 
At B and C they meet the same crossroad at distances AB = 4500 
yds., AC = 6450 yds. Two cyclists leave A at the same time along 
different roads; when they reach the crossroad they turn towards 
each other and meet just halfway between B and C 25 minutes after 
they left A . Find the speed of each cyclist. 

Ans. 11.04 miles and 8.38 miles per hr. respectively. 

13. In example 12, where would the cyclists have met had their 
speeds been the same? Ans. 671 yds. from C. 

14. A launch whose speed is 6 miles per hour sails in the direction 
N.W. A second launch having a speed of 8 miles per hour leaves at 
the same time from a point due south of the first launch and over- 
takes it in 5 hours. What was the course of the second launch, and 
how far apart were the launches when they started? 

15. Two streets OA and OB (Fig. 85) meet at an angle of 65 35'. 
The corner lot AOBC extends 142 feet along OA and 97 feet along 
OB. Find the dimensions AC and BC of the lot, AC and BC being 



i6o 



PLANE TRIGONOMETRY 



[chap. VIII 



at right angles to OA and OB respectively 
the lot. 

Suggestion. Solve (a) triangle AOB, (b) tri- 
angle ABC. To find the area we may apply 
formula (d), Problem 10, Exercise 33 (Why?). 

Ans. AC = 42.1 ft., BC = 111.9 ft., 
area = 8416.2 sq.ft. 



Find also the area of 




Fig. 85. 



o 



16. A, B, C are three mountain tops whose distances from one 

b another are known to be AB = 10.65 
miles = c, BC = 17.15 miles = a, CA = 
9.32 miles = b. From a point O at which 
C appears in a straight line with A, the 
horizontal angle COB measures 15 35' = 7. 



rJ±. 



c 
Fig. 86 




c) 



Find the distance BO = *Vs (s - a)(s - b)(s 

b sin 7 



17. From a point at the edge of a moat the angle of elevation of 
the top of a wall on the opposite side of the 
moat is 32 40' = a. Receding from the 
edge in a horizontal direction a distance of 
50 ft. = a, the angle of elevation becomes 
2i°35' = j3. Find the width x of the Fig. 87. 




moat, and the height y of the wall on the opposite side. 

a tan a tan 



A a tan 6 ,, 

Ans. x = = 80.5 ft., 

tan a — tan p 



y 



tan a — tan (3 



= 51.6 ft. 



18. From a point A at sea level, the angles of elevation a, a of 
P > two hills P, P' in the same straight line 
are a = 25 30' and o: / = 34 20' respec- 
tively. Walking towards the summits 
at an inclination 7 = io°, the angle of 
elevation was again observed at the 
moment that P' became hidden from 
view by P, and was found to be 
/3 = 42 15'. P is known to be h = 



■Mi- 




D 
Fig. 88. 



D' 



1595 ft. above sea level. Find the altitude of h' of P' . 

U = PW = PW u A&_ 
h " PB' AB' " PB' 



Suggestion. 



The law of sines will give 



76] SOLUTION OF OBLIQUE TRIANGLES 161 

P f 7? r A 7? r 

each of the ratios and in terms of known angles, hence 

AB PB 

7 , 7 sin ex sin (8 — a) r ; 

* = h ~ ' -T- yz 7T = 4372 ft. 

sm a sm (p — a ) 

19. The angular elevation of a standpipe S, from a place due 
east of it, is a = 35 27', and from a place 0' due south of and at a 

distance a = 200 ft. from it, the angular 

elevation is a! = 26 16'. Find the height h 

jJA of the standpipe, assuming the foot F of 

the standpipe and the points O and 0' to 



Jf ""^-^ 



\0 be in the same horizontal plane. 



--.a 



? ^J / Suggestion. 



Fig. 89. • From the triangle SFO, h = OF • tan a, 

From the triangle SFO', h = TTf ■ tan a, 
From the triangle OFO' ', (7f 2 - OF 2 = a 2 . 

Eliminate OF and OF' and solve for h. 

. -, a tan a tan a , ^ 

^4wj. h = - = 130.9 ft. 

V tan 2 a — tan 2 a:' 

20. From two points and 0' 200 ft. apart, a spire 5 subtends 
equal angles a = 21 i3 r . The bearings of 6* from and O r are W. 
and N. 25 W. respectively. Find the height h of the spire and its 
distance from 0, assuming that the points O and 0' are in the same 
horizontal plane with the foot of the spire. 

76. Applications from Physics. 

Exercise 39 

1. Two forces, one of 470 lbs., the other of 520 lbs., act on the 
same point at an angle of 37 27'. Find the resultant force. 

Ans. R = 938 lbs. 

2. If two equal forces meet at an angle 6, what angle will the 
resultant force make with either forc e and what w ill be the magni- 
tude of the resultant? Ans. J d, V 2 (1 + cos 0) times either force. 

3. A force of 100 lbs. is resolved into two components which make 
angles of 25°oo r and 35°oo / respectively with the direction of the 
original force. Find the magnitude of each component. 

Ans. 48.8, 66.2. 



162 



PLANE TRIGONOMETRY 



[chap. VIII 




4. A force of 500 dynes is resolved into two components, one of 
which is 200 dynes and makes an angle of 37 50' with the original 
force. Find the other component. 

5. Two forces of 15 lbs. and 25 lbs. respectively combine so as to 
form a resultant force of 30 lbs. Find the direction of the resultant 
with each of its components. Ans. 56 15.0', 29 55.6'. 

6. A ferry crosses a stream 360 yds. wide so as to make an angle 
of 1 2 5 with the direction of the current. 
The actual motion of the ferry is at an 
angle of 95 ° with the direction of the cur- 
rent. Find the actual distance the ferry 
made in crossing the river, and the distance A 

it would have made had there been no g * 9 ° 

current. Ans. 361.4 yds., 439.5 yds. 

7. Find the current of the river, Problem 6, and the speed of the 
ferry independent of the current if it required 5 minutes to cross 
the river. Ans. 1.24, 3 miles per hr. 

8. If in Problem 6 the speed of the ferry in still water is 5 miles 
per hour, find the time it would take to cross the river and the 
velocity of the current. 

9. A launch leaves the shore of a river running from west to east, 
and after two hours reaches an island a distance of 20 miles and in a 
direction due N.E. from the starting point. Assuming that the 
launch sailed in a straight line and with a speed which would have 
taken it 16 miles per hour in still water, find the velocity of the current. 

Ans. 10.8 miles or ^.^ miles per hour, according as the course of 
launch was west or east of north. 

10. A ray of light passes through a glass plate 12 mm. thick. The 
angle of incidence is 29 and the index of refraction is f. How far 
is it from the point where the ray leaves the plate 
to the point where it would have left it if it had 
suffered no refraction? Ans. 2.6 mm. 

11. From a point P at the bank of a river 
the angle of elevation of a flagstaff AB on the 
opposite side of the river as measured with a 
transit is a: = 47° 32', and the angle of depres- 
sion of the image of the flagstaff as seen in the 
water is 8 = 54 36'. The axis of the instrument Fig. 91. 




7 6] 



SOLUTION OF OBLIQUE TRIANGLES 



163 



is a = 4 ft. 6 in. from the level of the water. Find the height 
of the flagstaff. 

Suggestion. From the law of reflection of light, AB' = AB, hence 
from the triangle BDP, x — a = DP • tan a, 
and from the triangle B'DP, x + a = DP • tan 5. 

Eliminate DP and solve for x. Ans. 36.1 ft. 

12. Two billiard balls A, B are located at distances a = 2.53 and 

b = 4.13 respectively from the same cushion. 
Their distance apart is c = 5.84. At what 
angle must the first ball strike the cushion 
in order that on rebounding it may strike the 
second ball ? 

a + b 




Fig. 92. 



Ans. tan 6 = 



V( a -b + c)(- a+b+c) 



13. A, B, C are three points whose distances are AB = 25.0, 
BC = 20.0, CA = 10.0. What must be the inclination of a mirror 
at C to AB in order that a point of light at A may be reflected to B ? 

Ans. i3°34'.. 

14. In Problem 13, what will be the distance between A and its- 
reflection on AB or AB produced, if the mirror at C is inclined 
25 45' to the line AB ? 

15. A billiard table is p = 6.00 ft. long. Two balls are placed 
at P and P' whose distances from AD and AB are a = 2 ft. o in., 
b = 1 ft. 8 in., and a' = 4 ft. o in., b' = 
3 ft. 4 in, respectively. At what angle must ' 
the ball at P strike the cushion AB so that 
after it rebounds from the cushion BC it 
may hit P r ? Find also the distance of 
the points R and S from B at which the d 
ball strikes the cushion BC. 

b + b' 







R 


X 




l& ^ 






a 






p\ 




P 




a' 




\/ 


P' 



Ans. tan 6 = 



> x = (p — a) 



Fig. 93- 
b cot 6 = 2.00 ft., 



2 p — (0+ a') 
6 = 39 48' y = b'- (p - a') tan 6 = 1 ft. 8 in. 

16. A billiard table is p ft. long and q ft. wide. At what angle 
must a ball strike the long cushion in order that after striking each 
cushion in turn it may pass through its first position? 

Ans. = tan -1 ^. 
P 



164 



PLANE TRIGONOMETRY 



[chap, vni 




17. Two parallel forces p = 15, q = 25, act on levers of lengths 
a — 10, b = 3, respectively, on the com- 
mon fulcrum o. The force P makes an 
angle 6 = 30 with the lever a. At what 
angle </> must the levers be inclined in order 
to produce equilibrium ? Ans. <j) = 6o°. Fig. 94. 

77. Applications from Surveying and Engineering. 

Exercise 40 

1. In running a line AB an obstacle was encountered beyond B. 

The distance BP 200 ft. was measured, 
making the angle ABP = 154°= 6. At 
P an angle BPC = 65 ° = </> was turned 
off. Required the distance PC and the 
angle PCD in order that CD may be 




trie prolongation of AB. 
Ans. PC = 



BP • sin 6 



= 87.7, angle PCD = 91 . 




sin (6 - 0) 

2. In surveying a boundary A BCD an obstacle was met between 

-S and C. Instead of measuring BC directly, 

BP =255 ft. was measured, making the 

angle ABP = 95 3o / . Then PC = 234.5 ft. 

was measured, making the angle BPC = 

104 34'. Required the distance BC and 

the angle ABC. 

Ans. BC = 387.4 ft., angle ABC = 131 21.7'. 

3. In running a traverse A BCD an impenetrable thicket is en- 
countered between B and C. From B 
a road extends to P, where it is met 
by a crossroad which leaves the thicket 
at Q. The following measures were 

D ' taken: 

Angle ABP = 35 45', 
BP = 674 ft., 
angle5P<2=iio°37', 

PQ == 890 ft., angle PQC = 115 38', QC = 345 4 ft., angle 
QCD = 33 23'. Required the distance BC and the angles ^4I>C and 
BCD. 

Ans. BC=i3i6 ft., angle ABC = gi° 05', angle £CX> = in° 48'. 







77] 



SOLUTION OF OBLIQUE TRIANGLES 



165 




B is a point, invisible and inaccessible from A, whose distance 
from A was to be found. From A a line 
was run to two points P and Q from each 
of which B was visible, and the follow- 
ing measures were taken: AP = 236.7 ft., 
angle APB = 142 37.3', PQ =215.9 ft-! 
angle AQB = 76 13. 8'. Required AB. 
Fig- 98- Ans. AB = 441.0 ft. 

5. A surveyor wished to find the distance of an inaccessible point 
from each of two points A and B, but had 
no instrument with which to measure angles. 
He measured A A' = 150 ft. in a straight line 
with OA and BB' =250 ft. in a straight line 
with OB. He then measured AB = 279.5 ft., 
BA' = 315.8 ft., A'B' = 498.7 ft. Required 
AO and BO. 




Fig. 99. 




6. Two inaccessible points A and B were visible from C, but no 
a n other point could be found from which both 

A and B were visible. A point P was se- 
lected from which both A and C were vis- 
ible. CP = 425.3 ft., angle APC = 37 15.4', 
q angle ^4CP = 42°35.3 / . A second point Q 
Fi S- IO °- was selected from which both B and C were 

visible. CQ = 405.4 ft., angle BQC = 53 14.8', angle BCQ = 58 04.7', 
angle ACB = 65 10.5'. Required AB. 

Ans. AB = 336.8 ft. 

7. It was required to find the distance between two inaccessible 
points A and B. No point could be found 
from which both A and B were visible. Two 
points P and Q were selected from which A 
could be seen, and two other points R and S <f 
from which B could be seen. The following 
measures were taken: 

PQ = 200 ft., angle PQA = 6i° 30', angle QPA 

RS = 250 ft., angle RSB = 29 16', angle 57?5 = 72 10', 

PR = 300 ft., angle PRS = 121 36', angle RPA = 92 54'. 

Find AB. 




7o° 25', 



i66 



PLANE TRIGONOMETRY 



[chap. VIII 



Suggestion, (a) In triangle QPA, find PA; (b) In triangle APR, 
find AR; (c) In triangle RSB, find ££; (J) In triangle ARB, find 45. 
To check, compute AB again using the triangle APB. 

Ans. AB = 480 ft. 

8. A circular tower subtends a horizontal angle of 2 a = 20 13' 
at a distance of a = 198 ft. from the edge of the tower, measured on 
a diameter produced. Find the diameter of the tower. 

2 a sin a 



Ans. Diameter = 



= 84.3 ft. 




Fig. 102. 



1 -sina 

9. Three stations A, B, C are to be connected by a juncture O, 
equally distant from each of them. The distance of A from B is 
25.3 miles; of B from C, 17.8 miles; and of C from A, 19.4 miles. 
Find the distance of O from A. Ans. 12. 7 miles. 

10. Fig. 102 represents a cross-section map. The horizontal 
scale is 1 in. = 100 ft., the vertical 

scale is 1 in. = 10 ft. AB = 5 in., 
BC = 6.35 in., CD = 4.13 in., BB' = 
4.98 in., CC = 1.76 in., DD' = 3.87 
in. A and D' are to be connected 
by a street of uniform grade. Re- 
quired the inclination of AD' to AD, the cut B'B" that must be 
made at B' and the fill CC" at C . 

11. On a map drawn to a scale 1 : 5000, three points A, B, C are 
tB > at distances AB = 5.03 in., BC = 4.23 in., CA = 

3.54 m. The points A', B', C of which A, B, C 
are the projections, lie 357.25, 713.27, and 623.53 ft. 
respectively above an arbitrarily chosen hori- 
B zontal plane, called the datum plane. Required 
the actual lengths of the lines A'B\ B'C ', C A' , 
the angles formed by these lines, and their in- 
clinations to the datum plane. 

Ans. A'B' = 2540.1 ft., 8° 03' 26", C'A'B' = 55 18' 56". 
C'B' = 2116.9 ft., 2 25 r 46", A' B'C = 44 o2 x 58". 
A'C = 1789.9 ft., 8° 33' 20", B'C A' = 8o° 38' 14". 

12. From a hilltop H two landmarks A and B in the valley, 
which are known to be a mile apart, subtend a horizontal angle 
Q = 105 34.4' and their angles of depression are a = 27 27.5' and 




77] 



SOLUTION OF OBLIQUE TRIANGLES 



167 



fi = 27 54. 9' respectively. Find the height h of the hill above the 




valley. 

Suggestion. 
b respectively 

a = h cot /?, 



Denote OB and OA by a and 
Then 



b = h cot 



Of, 



5 



Fig. 104. 
tan § (4 - 5) = 



and by the law of tangents 
A cot (3 — h cot a 



tan %(A+B) 



h cot |8 + h cot a: 

tan a — tan 3 , , -, M 

= ; 777 — a * tan (90 ~ i 0), 

tan a + tan p 

from which A = angle 0^4 B and 2? = angle OB A may be found. 

Then 

sin A 



a = AB 



sin 



, and finally h = a tan p\ 



^4^5. /? = 2043 ft. 

13. From a mountain top 3257 ft. high, two hotels are observed 
whose altitudes are 627 and 937 ft. respectively. The angles of 
depression of the hotels are 45 34' and 58 27' respectively. Find 
the distance (horizontal) between the hotels. 

14. A tower ED is situated on the side of a hill whose summit C 
is 550 ft. above BA. At a point B from 
which the top of the tower and the top of 
the hill appear in a straight line, the angles 
of elevation of the foot and top of the 
tower are 15 30' and 29 27' respectively. 
The distance of B from the foot of the 
hill at A is 600 ft. Find the height of the tower and its distance 
from the foot of the hill. Ans. ED = 212 ft., AD = 248 ft. 

15. To determine the height H of a building above the steps, the 

horizontal distance d between two stations 
A' and A" was measured with a tape, the 
heights h! and h" of these stations were de- 
termined by level readings on the top of 
the steps, and the angles of elevation 6' and 
O" of the top of the building as seen from 
A' and A" were measured by means of a 

Fig. 106. transit. Show that 





i68 



PLANE TRIGONOMETRY 



[chap. VIII 



H = 



h" tan B'- h! tan B" - d tan B' tan B' 
tan B' - tan B" 



1 6. To determine the area of a triangular field, the sides were 
measured with a ioo ft. steel tape, which was afterwards found to 
be 0.3 ft. too long. The sides as measured were 276.33 ft., 391.81 ft. 
and 318.00 ft. Determine the area as measured and also the cor- 
rected area. 

Ans. Measured area = 43,527 sq. ft., 

Corrected area = 4 ^ ' — i = 43,789 sq. ft. 

ioo 2 

What would have been the corrected area if the tape had been found 
0.3 ft. too short? 



17 




In running a 7 curve (radius 819.02 ft.) an obstacle was 
B encountered beyond the point A. In order to 
locate a point P beyond the obstacle, the tangent 
at A was produced to B, a distance of 500 ft. At 
B an angle ABC was turned off equal to 37 . 
Required the distance BP. 

Suggestion. From the right triangle OAB, find 
OB and the angle OB A, then use the triangle OPB 
to find BP. Ans. BP = 153.28 ft. 

The equations give a second solution. Interpret it. 

18. The inaccessible distance PQ = x may a a 

be found by measuring PA = a, QB = b, in 
a straight line with PQ, and the angles a, /5, 7 
which a, b, and % subtend at some accessible 
point 0. Show that x is given by the equa- 
tion 

(a + x) (b + x) _ ab 
sin (a + 7) sin (J3 + 7) sin a sin /3 

Suggestion. Find the ratio PO : QO, first from the triangles OAP 
and OAQ and second from the triangles OBQ and OBP, and compare 
the results. 

19. From a boat P over a submerged rock, a sextant is used to 
measure the angles subtended by three known points A, B and C on 
shore. The readings give APB = 57 13.2', BPC = 65 21.3'. From 




77] 



SOLUTION OF OBLIQUE TRIANGLES 



169 




a previous survey it is known that angle ABC = 116 53. 6', AB = 
1846 ft., and BC = 1493 ft. Find the distances AP, BP and CP. 
(This is another illustration of the three-point problem, the three 
points this time being given by two distances and an angle.) 
Ans. AP = 2137 ft., BP = 1582 ft., 

r~< t» c z± A Ames 

CP =IOOI ft. A A Blair 



20. In the quadrilateral Ames-Blair 
Cole-Davis, of a triangulation system, the 
angles were measured and adjusted with A Davis 
the following results: 

At Ames 

To Blair 

To Cole 48° 53' 31", 

To Davis 109 12' 37". 

At Cole 
To Davis oo° oo' 00", 
To Ames 49 13' 02", 
To Blair ii9°37'34". 



~~° ~~' ~~>> 
00 00 00 , 



1 A Davis 


Fig. 109. 


V' 

A Cole 


At Blair 




To Cole 


oo° od 


OO , 


To Davis 


3i° i5' 


59", 


To Ames 


6o° 41' 


57". 


At Davis 




To Ames 


oo° oo' 


00", 


To Blair 


41° 21' 


25", 


To Cole 


70 27' 


52". 



The distance Ames-Blair measured 12,362 ft. Find all the other 
lines in the figure. 

Ans. Blair-Cole = 9,887 ft., Ames-Cole = 11,443 ft-? 
Davis-Cole = 10,549 ft., Ames-Davis = 9,194 ft., 
Davis-Blair = 17,667 ft. 



21. After adjustment the angles of the adjacent triangulation 
figure were found to be as follows: 



2^"1119 -^ 



Mo 1 . 




(1) 37°44 , i6", 

(2) 63° 21' 53-, 

(3) 78° 53' 51", 

(4) 93°4i / 23", 

(5) 25° 18' 17", 

(6) 6i°oo r 20", 



(7) 57°3i , o 3 ", 

(8) 68° io r 11", 

(9) 54° 18' 46", 

(10) 35° 08' 11", 

(11) ioi°53 , o6 ,/ , 

(12) 42° 58' 43". 



The side AB measured 4403.2 ft. Find EF 
EF 



AB * sin (2) sin (4) sin (8) sin (n) = 75o7>6 ft# 
sin (3) sin (6) sin (9) sin (12) 



170 



PLANE TRIGONOMETRY 



[chap. VIII 



22. It is required to find the sides of the triangle EFG, from the 



following adjusted results furnished by a 
survey: 

€1=54° 08' 07", 

o / // 

€2 = 151 20 II , 
*z = 65° 54' 59", 



a = 40° 31' 15", 
ft = 65 15' 16", 
ft = 54° 5°' So", 

O / // 

71 = 59 10 02 , 

72 = 74° 13' 27", 










m = 25 25 25 ', 




772 = 80° 34' 10". 
The base line AB = 76,542 ft. 

EF = AB ' s * n a s * n ^ 2 s * n T2 s * n ^ 2 ""^ ^ = 72^1 ft 
sin (a + j8i) sin (0 2 + 71) sin ei sin 171 

77^ -E^ * sm ^2 r + 77^ EF • sin e 3 r , 

EG= . - = 12,920 ft., FG= -7— — —*-= 11,957 ft. 

sm 1,772 + e 3 ) sm (772 + €3) 

23. The sides b and d and the area of a quadrilateral field are to 
be computed, the following data having been 
determined by measurement. a = 403.9 ft., 
C = 200.9 f t., a = 115° 12.3', ft = 40° 25.3', 
& = 42°37-5', J = I2i°i2.i / . 
Check the result for d. 



d/ 



■ib 



Fig. 112. 



Area — \ a 



sin a 
sin /3i 



[a sin (a + 0i) + c sin 2 ] = 85,308 sq. ft. 



24. A surveyor measured the following distances and their re- 
spective bearings: 

AB = 413 ft., bearing N. 72 15' E. 
BC = 395 ft., bearing N. 37 27' W. 
CD = 525 ft., bearing S. 68° 13' W. 

What distance and bearing will carry him to 
the starting point? 

Suggestion. The distance DA and its bear- 
ing are most easily computed from the right triangle DAS. 
AS = AP + QC-RC = AB cos 72 i$'+BC cos 37 2? -DC cos 68° 13', 
SD = PB - QB-DR = AB sin 72 15'- BC sin 37 27' '-CD sin 68° 13' '. 
Ans. DA = 688.8 ft. Bearing S. 69 01. 6' E. 

AP, QC and CR are called the latitudes, PB, BQ and RD the depar- 
tures of the courses AB, BC and CD respectively. The latitudes are 
considered positive or negative according as the course bears north 




78] 



SOLUTION OF OBLIQUE TRIANGLES 



171 



or south, the departures positive or negative according as the course 
bears east or west. In the example given, AS is positive, SD nega- 
tive, hence the bearing of AD is N.W. and the bearing of DA is S.E. 
25. After adjusting the field measurements in a traverse survey 
around an area, the following bearings and distances were obtained: 



Stations. 


Bearing. 


Distance. 


Latitude. 


Departure. 


1-2 

2-3 
3-4 
4-5 

5-6 
6-1 


N. 44° 38.5' E. 
N. 52° 42.5' E. 
S. 2° 00.0' W. 
N. 88° 34.0' W. 
S. 88° 50.5' W. 
N. oo° 00.0' 


287.3 
451.8 
921.3 
212 .0 

443-7 


+ 204.4 

+ 273-7 

— 920.7 

+ 5-3° 

— 6.41 

+ 443-7 


+ 201.9 
+ 359-4 

- 32-I5 

— 211. 9 

-317.2 
000.0 



Compute the latitudes and departures given in the columns above. 
Draw a figure and compute the area. 

78. Applications from Navigation. 

Exercise 41 

1. A cape as seen from board ship bears N. by E. The ship sails 
N.W. 35 miles and then the cape bears exactly east. How far is 
the cape from the second point of observation ? Ans. 29.7 miles. 

2. To determine the distance between two ships at sea, an 
observer noted the interval of time between the flash and report of 
a gun fired on board each ship, and measured the angle which the 
two ships subtended. The intervals were 5 and 7 seconds respec- 
tively, and the angle was 37°45'. Assuming the velocity of sound as 
1090 ft. per second, find the distance between the ships. 

Ans. 4707 ft. 
Discuss the accuracy of your result. 

3. From a ship two rocks are seen in a straight line bearing 
S. 22°3o / E. After the ship sailed 8 miles S.E. the first rock bore 
N. 65 13' W. and the second N. 76°46 / W. Find the distance be- 
tween the rocks. 

Ans , 8.sm22°ysinri°33' =iiimae5| 
sin 42 43 sin 54 16 

4. A privateer, 8 miles S.W. of a harbor, sees a ship leave the har- 
bor. The course of the ship is S.E by S. and its speed 10 miles. In 



172 PLANE TRIGONOMETRY [chap, vnr 

what direction and at what speed must the privateer sail in order to 
overtake the ship in 2 hrs. ? 

5. A ship sails N.N.E. at a rate of 12 mi. per hr. A second ship 
making 15 miles per hour starts at the same time from a point 50 miles 
east of the first ship. In what direction must the second ship sail 
in order to meet the first ship, and how long will it take before they 
meet? Ans. N. 55 $$' W. 2 hr. 57 min. 

6. A ship sailing east sees two lighthouses 15 miles apart in a 
line E.N.E. After sailing an hour, one lighthouse bears N. by W. 
and the other N.E. by N. Find the speed of the ship. 

7. From the top of a mast 70 ft. above the level of the sea, the 
light of a distant lighthouse is seen just above the horizon, and after 
the ship has sailed directly toward the lighthouse for 45 min. the 
lighthouse becomes visible from the deck of the ship, which is 12 ft. 
above the level of the sea. Find the rate at which the ship is moving, 
considering that the earth is a sphere 8000 miles in diameter. 

Ans. 8 miles per hour. 

8. A ship leaves Cape Flattery (48 23' N. Lat., i24°44 r W. Long.) 
and sails W. 100 miles, N. by E. 150 miles, S.E. 113 miles. Find the 
position reached. Ans. 49 io' N. Lat., 124 30' W. Long. 

Suggestion. 1 mi. = 1' of arc. 

79. Problems from Astronomy and Meteorology. In the 

following problems the planets are assumed to move in the ecliptic 
and their orbits are taken to be circles with the sun at the center. 

Exercise 42 

1. The mean distance of Venus from the sun is 36,000,000 miles and 
that of the earth from the sun 92,900,000 miles. How far is Venus 
from the earth when its angular distance from the sun is 13 35'? 

Ans. Either 61,700,000 or 119,000,000 miles. 

2. The mean distance of Mars from the sun is 141,500,000 mi., 
What is the angular distance between Mars and the sun when the 
distance between Mars and the earth is 122,400,000 miles? 

3. What is the distance between Jupiter and the earth, when its 
angular distance from the sun is I23 i2 r , the heliocentric longitude 
of the earth being 67 59' and that of Jupiter 21 23'? 

Ans. 381,000,000 miles. 



79] 



SOLUTION OF OBLIQUE TRIANGLES 



J 73 



4. The mean distance of Saturn from the sun is 886,000,000 miles 
that of Uranus is 1,781,900,000 miles. What is the difference in the 
heliocentric longitude of these planets if their angular distance is 

47° 45'? 

5. From the top of a mountain a miles high the corrected angular 
depression of a line tangent to the earth's surface is found to be 6. 
Show that the diameter of the earth is 



d = 



2 a cos 9 
1 — cos 



6. Two observers A and B observe the same meteor. As seen by 
A, the altitude a = 6i° 35', the azimuth £1 = 119 56'. As seen by B, 
who is stationed 18.5 miles north of A, the azimuth of the meteor is 
£2 = 85°32 / . Compute the height h of the meteor and its altitude 
a as seen by B. 



fe = asmr 2 tana =62>Q mi les> a'= tan^ sin ^ 2 tana = 25 
sin (f 1 + f 2 ) sin U 



°n'. 



7. An observer measures the angle of elevation of a sharply 
defined cloud which is in the same vertical plane with the sun. The 
angle measures 48. 9 . The altitude of the sun at the same time 
measures 56. 6°. The shadow of the cloud was 875 ft. from the 
observer. Find the height of the cloud. Ans. 41 10 ft. 

8. From two points and 0' on the same meridian, the zenith 
distances ZOM = 35 25' 20", Z'O'M = 40 io r 50", 
of the moon are measured. The difference in 
latitude between the points of observation is 
ZCZ' = 74 26' 18". Find the distance of the 
moon from the earth, assuming the radius of the 
earth as 3958.8 miles. 

Fig. 114. Ans. 238,840 miles. 

9. On a clear day, twilight ceases when the sun has reached a 

position 1 8° below the horizon (HAS = 18 ). Find 

the height AE of the atmosphere which is sufficiently 

dense to reflect the sun's rays. Take OC = 4000 

miles. The result must be diminished by 20 % to 

allow for refraction. 

Ans. 40 miles. Fig. 115. 





174 PLANE TRIGONOMETRY [chap, vin 

80. Geometrical Applications. 

Exercise 43 

1. Two sides and the included angle of a parallelogram are 156, 
237 and 37 50' respectively. Find the length of the diagonals and 
the area of the parallelogram. 

Ans. Diagonals, 148.7, 372.7; area, 22,680. 

2. Find the area of a regular pentagon whose side is 12 ft. 

Ans. 274.74 sq. ft. 

3. The sides of a triangle are 17, 20, 27 respectively. Find the 
length of the median to the longest side. 

4. In Problem 3, find the length of the bisector of the greatest 
angle. 

5. Given a side c and two adjacent angles A and B of a triangle. 

Find expressions for the perpendiculars p A , p B , p c drawn from the 

vertices A, B, C to the opposite sides. 

a j. • t> - a c sin A sin B 

Ans. p A = csmB, p B =csmA, p c = . • 

sin (A + B) 

6. Find the area of the segment of a circle whose radius is 15.6 
and the base of the segment 21.3. 

7. Find the length of the diagonals of a regular pentagon whose 
side is 10. Ans. 16.18. 

8. Find the lengths of the diagonals of a regular heptagon whose 
side is 12. Ans. 26.96, 21.62. 

9. Given the perimeter 25 and two angles of a triangle 55 and 65 
respectively, to find the sides to three places of decimals. 

Ans. 7.902, 8.743, 8.355. 

10. Given one side a of a triangle, the sum / of the other two sides 
and the angle A opposite to the first side, to find the angles of the 
triangle. 

Ans. cos \ (B- C) = - sin \ A, \ (B + C) = 90 - i A, 

a 

from which B and C may be found. 

n. Two sides of a triangle are 13.5 and 17.6 respectively and the 
included angle is 35 16'. Find the side of an equilateral triangle 
having the same area. Ans. 6.293. 



80] SOLUTION OF OBLIQUE TRIANGLES 1 75 

12. In Problem n, find the side of an equilateral triangle having 
the same perimeter. Ans. 13.77. 

13. The area of a regular polygon inscribed in a circle is to the area 
of a regular polygon of the same number of sides circumscribed 
about the circle as 3 is to 4. Find the number of sides of the 
polygon. 

14. Two parallel chords on the same side of the center subtend 
angles of ioo° and 50 respectively. Find the distance between the 
chords when the radius is 15.34. Ans. 4.042. 

15. A regular pentagon and a regular decagon have equal perim- 
eters. Find the ratios of their areas. Find the area of each, if the 
sum of their areas is 536 sq. in. 

Ans. Ratio, 1.0487; area of pentagon, 274.36; 

area of decagon, 261.64. 

16. The coordinates of the vertices of a triangle are (o, o) 
(471.32, o), (896.41, 230.00) respectively. Find the coordinates of 
the intersection point of the bisectors of the angles. 

Ans. (456-7, 57-66). 

17. The side of a regular inscribed heptagon is approximately 
equal to one-half the side of an inscribed equilateral triangle. 
Find the error of the angle at the center, subtended by one of 
the sides as determined by this rule. Find the error in the length 
of the side of the heptagon if the radius of the circumscribed 
circle is 100. 

18. The following rule gives approximately the side of a regular 
inscribed nine-sided polygon. On the diameter i5 as a base 
(Fig. 116) construct a triangle ABC, having for 
the other two sides BC and AC, the sides of a 
regular inscribed hexagon and square respectively. 
From A as a center and a radius equal to AC, 
draw an arc cutting the diameter in D. CD will 
be approximately equal to the side of a regular Fig. 116. 
nine-sided polygon. Determined the error in the central angle of 
the polygon as determined by this rule, and the error in the length 
of the side. 




176 PLANE TRIGONOMETRY [chap, vm 

Suggestion. Apply the law of cosines to determine the angle A, then 

CD 

CD = 2 AC cos § A, and the angle at the center AOE = 2 cos l — ■> 

2 r 

where r is the radius of the circle. 

Ans. Error in central angle = o°94 / , 

Error in length of side = 0.00254 X r. 

19. The following rule gives approximately the side of a regular 

inscribed eleven-sided polygon. Let A, B, C, D be four consecutive 

c^^=====^jb vertices of a regular inscribed hexagon. Bisect 

/f\\^y \\p t ne diagonal BD in E. Draw AE. From C as 

f/^''E~~^jp/*\ a center and radius equal to a side of the hex- 

J^ ^— — ^^^jj agon, draw an arc cutting AE in F. EF will 

be approximately equal to a side of the regular 
inscribed eleven- side. Compute the error in the 
central angle and in the side of the polygon thus determined. 

Suggestion. ABE is a right triangle, the two legs of which are 
readily found in terms of the radius r. Hence angle A EB can be 
found. Then in the triangle CEF, angle CEF = 90 + angle AEB f 
and the two sides CE and CF can be found, hence EF can be 
computed. 



CHAPTER IX 
* THE GENERAL ANGLE AND ITS MEASURES 

81. General Definition of an Angle. In elementary geometry an 
angle is defined as the difference in direction, or the amount of open- 
ing, between two lines which meet or tend to meet in a point. Obvi- 
ously, two lines differ most in direction, the opening between them 
is greatest, when they extend in opposite directions. According to 
this definition, no angle can be greater than a straight angle, and in 
fact every angle considered in plane geometry is less than two right 
angles. 

For the purposes of trigonometry and higher mathematics in gen- 
eral it is convenient to think of angles as formed by revolving a 
line in a plane about a fixed point. The line in its first or initial 
position is called the initial line, in its final position it is called the 
terminal line, the fixed point is called the vertex. The amount of 
rotation which brings the line from its initial position to its terminal 
position is called the angle between the two lines. 

With this definition, it is plain that an angle may have any magni- 
tude whatever, for their is no limit to the amount of rotation which 
a line may undergo. The angle described by the big hand of a clock 
increases so long as the clock continues to run, — when it has run 15 
minutes the angle described is a right angle ; when it has run 30 min- 
utes the angle described is a straight angle; after an hour the big 
hand has returned to its initial position, yet the angle which it has 
described is not zero but four right angles. As the hand continues 
to move, the angle between the hand and its initial position still 
increases. After two complete revolutions this angle is equal to 
eight right angles; after three complete revolutions, to 
twelve right angles, and so on. 

Before we can assign a magnitude to an angle in 
this wider sense, we must know more about it than 
merely the position of its sides. By the angle AOB Fig- "8. 
(Fig. 118) may be meant an angle of 45 , as indicated by the short 
arrow, or an angle of 405 , as indicated by the long arrow, or 

177 





1 78 PLANE TRIGONOMETRY ' [chap, ix 

an angle of 45 increased by any number times 360 , depending 
on how many complete revolutions OB has described in reaching 
its final position. The angles 45 , 405°, and all the other angles 
which AOB might represent, are said to be coterminal. In ele- 
mentary geometry, and generally when nothing to the contrary is 
said or implied, in referring to an angle, the numerically smallest of 
all the coterminal angles is understood. This value is known as the 
principal value of the angle. 

82. Positive and Negative Angles. Sometimes it is convenient 
to distinguish between the two directions in which rotation of a line 

about a point in a plane may take place. This 
is done by prefixing a plus sign to the resulting 
angle if it has been described by rotation in a 
Fig. 119. direction contrary to that in which the hands of 

a clock move (counterclockwise), Fig. 119, and a 
minus sign if the rotation has been in the op- 
posite direction (clockwise), Fig. 120. When the 
sign is considered, the initial line is always read 
first; thus, each of the angles in figures 119 and 
angle AOB." 

83. Complement and Supplement. If the algebraic sum of two 
angles is equal to a right angle, each angle is said to be the comple- 
ment of the other; if their algebraic sum equals two right angles each 
is said to be the supplement of the other. That is to say, — 

If A + B = 90 , A and B are complementary angles, 

If A + B = 180 , A and B are supplementary angles. J 

A or B may have any magnitude, positive or negative. Thus, since 
95 + (— 5 ) = 90 , 95 and — 5 are complementary angles, 

and since 

— no°+ 290°= 180 , — no° and 290 are supplementary angles. 

84. Angles in the Four Quadrants. Let the vertex of the angle 
be taken as the origin of a system of rectangular coordinate axes, 
and the initial side of the angle for the positive direction of the 
#-axis. It follows that every angle less than 90 , as angle AOP^ 





85] THE GENERAL ANGLE AND ITS MEASURES 179 

Fig. 121, has its terminal side in the first quadrant, and for this 

reason it is said to be an angle of the first quadrant. Every angle 

which is greater than a right angle but less than 

a straight angle, as angle AOP 2 , is said to lie in, 

or to be an angle of, the second quadrant. 

Similarly, the angle AOPz is an angle of the 

third quadrant, angle AOP4 an angle of the 

fourth quadrant, but an angle greater than four 

right angles and less than five right angles falls 

again in the first quadrant, and so on. Negative . 

angles also lie in the quadrants determined by 

their terminal sides. Thus, while angle AOB (Fig. 119) lies in the 

first quadrant, the angle AOB (Fig. 120) lies in the fourth quadrant. 

85. Sexagesimal Measure of Angles. It is familiar to all that 
different units of measures may be employed in the measurement of 
a given magnitude. Thus, any distance may be expressed in inches, 
feet, yards, rods and miles of the English system, or in millimeters, 
centimeters, meters and kilometers of the metric system. Values 
may be expressed in dollars and cents, in pounds and shillings, in 
francs and centimes, etc. Just so, angular magnitudes may be 
expressed in terms of different units of measure. In order to ex- 
press a measure given in one set of units in terms of the units of 
another system, it is only necessary to know the relation between 
the units of the two systems. 

The system of angular measure which we have used thus far is 
called the sexagesimal system* The total angular magnitude about 

* From the Latin word sexagesimus (sexa, six + decimus, one-tenth) , meaning 
one-sixtieth. 

The sexagesimal scale was once applied to many measures. The sixtieth part 
of a unit of time, and length and weight was called its primate or prime; each 
prime was divided into sixty seconds, each second into sixty thirds. The divisor 
60 has now disappeared among Western nations, except in measures of angles 
and of time. It is probable that the division of the angular space about 
a point into 360 degrees originated with the Babylonians, whose year consisted 
of 360 days. The Latin word for degree is gradus, the Greek, bathmos, each of 
which means step; that is, a degree originally meant the daily step of the sun 
eastward among the stars. The Chinese knew many centuries ago that the year 
consisted more nearly of 365! days, so they divided the total angular space about 
a point into 365 \ degrees. The sexagesimal system is unscientific and is doomed 
to be replaced sooner or later by a better system. 



180 PLANE TRIGONOMETRY [chap, ix 

a point is divided into four equal parts and each part is called 
a right angle. The ninetieth part of a right angle is called a 
degree (°), the sixtieth part of a degree is called a minute ('), and 
the sixtieth part of a minute is called a second ("). The sexa- 
gesimal system of angular measures is the one most frequently used 
in every- day life. 

86. Decimal Division of Degrees. Instead of subdividing into 
minutes and seconds, the degree is sometimes divided decimally, 
into tenths, hundredths and thousandths. The decimal division of 
the degree has been used more or less ever. since the invention of 
decimal fractions in the sixteenth century. Tables based on the 
decimal division of the degree have been published at various 
times.* 

87. Centesimal Measure of Angles. Another system of angular 
measure, known as the centesimal, is obtained by dividing the right 
angle into ioo equal parts, each of which is called a grade (g), each 
grade into ioo minutes ( v ), and each minute into ioo seconds ( vv ). 
An angle of 25 grades 16 minutes and 78 seconds would be written 

25^ i6" 78 vv , or more simply, 25.1678^. 

The centesimal system of angular measures was introduced as a 
part of the metric system by the French reformersf at the time of 
the great revolution. It possesses many advantages over the. sexa- 
gesimal system, but owing to the fact that nearly all reference 
books and tables, all records of observation, the graduation of all 
astronomical instruments and of most engineering instruments, as 
well as the scales of geographical and nautical maps and charts, are 
based on the sexagesimal system, the progress of the introduction 
has been slow. But in spite of the many obstacles to be overcome, 
the system is making steady gains in the countries where the metric 
system is used. The centesimal system is now used exclusively in 
the field surveys in France, Belgium, Hessia and Baden, and it is 
legally recognized in several other states. It is regularly taught in 
many European high schools and technical schools. 

* The decimal system is now taught alongside the sexagesimal system in 
Harvard University and a number of other Eastern institutions of learning. 

f The invention of this system and the first attempt to introduce it dates back 
to 1783 and is due to a German by the name of Johann Karl Schultze. 




88] THE GENERAL ANGLE AND ITS MEASURES 181 

88. The Circular or Natural System of Angular Measures. In 

many practical investigations and in nearly all theoretical work, it 
is convenient to employ what is known as the circular or natural 
system, of angular measure. 

It is shown in geometry that in concentric circles the arcs a and 
a' (Fig. 122) intercepted by any angle a at the center are propor- 
tional to the radii r and r' of the circles, that is, 

a a' , N 

-== — > (1) 

r r 

and again, that in the same circle two central 
angles a and (3 are to each other as their inter- 
cepted arcs a and b, that is, 

Fig. 122. « = 9l = <£ Ik r 9 \ 

fi b r/r K} 

From (1) it follows that the ratio of the length of the arc to the 
radius of the circle is independent of the length of the radius, ■ — 
in other words, that this ratio is constant so long as the angle is 
constant — and from (2) that this ratio varies as the angle, and 
may therefore be used as the measure of the angle. This ratio is 
known as the circular measure of the angle. 

The circular measure of an angle is the ratio of the length of its inter- 
cepted arc, in a circle whose center is at the vertex of the angle, to the 
radius of the circle. 

The unit of circular measure or natural unit is obtained by making 
a equal to r, that is, by taking the angle such that the intercepted 
arc equals the radius. This unit is called a radian. 

A radian is an angle which, when placed with its vertex at the center 
of a circle, intercepts an arc equal in length to the 
radius of the circle. 

Thus, if the arc AB (Fig. 123) is equal in length / 
to the radius OA, the angle AOB measures one \ ' radius 

radian. v \ / 

One peculiarity of the circular system is that it ^ — ""* 

has no subsidiary or derived units ; that is, no 
other units which are multiples or sub-multiples of a radian. All 
angles large or small are expressed in terms of this single unit, the 
radian. 




182 PLANE TRIGONOMETRY [chap. 



IX 



89. Comparison of Sexagesimal and Circular Measure. To 

find the relation between the two kinds of units, degrees and radians, 
it is best to compare the two measures for the entire angular space 
about a point, that is, of four right angles. Expressed in circular 
measure, the measure of this angular space is equal to the circum- 
ference of a circle divided by the radius. Now the circumference of 
a circle is equal to the diameter, or twice the radius, multiplied by 
3.14159 + . Denoting this number by the Greek letter ir* we have 
for the circular measure of four right angles 

circumference 

■ = 2 7T. 

radius 

Measured in degrees the same angular space is 360 , hence we 
obtain the fundamental relation 

2 7r radians = 360 degrees, 
or ir radians = 180 degrees. (1) 

This relation enables us to reduce radians to degrees, and vice 
versa. Dividing both sides of (1) by the number w, we obtain 

1 radian = degrees, 

7T 

hence, — 

* The number ir is a most marvelous number. It is incommensurable, that is, 
it cannot be exactly expressed by a fraction whose terms are whole numbers. 
Nor can it be found by taking the square root, cube root or higher root of some 
•commensurable number. Neither is it the root of any algebraic equation. For 
this reason it is called a transcendental number. 

However, the value of -k may be computed to any desired degree of accuracy. 
Archimedes showed that it is less than 3} and greater than 3yy. The former value 
is still commonly employed in rough approximations. 

The Hindus, as early as the sixth century, computed the value of ir from the 
perimeter of a regular inscribed polygon of 384 sides and found ir = 3.1416; 
the value now generally used where 37- = 3.1428 is not sufficiently accurate. 

In recent times, the value of ir has been computed to 707 places of decimals. 
The first 32 places were computed by Ludolph van Ceulen, a Dutchman, who 
devoted a good portion of his life to this task. For this reason ir is frequently 
referred to as Ludolph's number. Its first 30 places are 

7T = 3.141,592,653,589,793,238,462,643,383,279. 



89] THE GENERAL ANGLE AND ITS MEASURES 183 

T o_ 

To reduce radians to degrees, multiply the number of radians by -— 

7T 

(equals 57.3 nearly*). 

Again, dividing both sides of (1) by 180 and writing the second 
member first, we obtain 



1 degree = radians; 

180 



hence, - 



T 



To reduce degrees to radians, multiply the number of degrees by 

180 

(equals 0.0175 nearly). 

Example i . How many degrees in a — radians? 

2 

Solution. Since w radians = 180 , -radians = 90 , 

2 

and — radians = 270°. 
2 

Example 2. How many radians in an angle of 6o°? 

Solution. Since 180 = x radians, 6o° = - radians. 

3 

It is best to retain the letter x in the answers unless there is some 
reason to the contrary. For instance, putting for -k its value 3.1416 
and dividing by 3, the answer to the second example might have 
been written 6o° = 1 .047 2 radians, but 71-/3 radians is preferable. Like- 
wise the first example might have been stated thus : How many de- 
grees in 4.7144 radians? but the statement as first given is preferable. 

When a number represents the measure of an angle, and no unit 
is expressed, the natural unit is understood. Thus, when we speak 
of the angle 71-/2, we mean not 71-/2 degrees but 71-/2 radians. The 
angle -k means w radians or 180 , the angle 2 means not 2 but 2 radi- 
ans, or 114.5916 . We may look upon ^7r, 7r, 2 7r, 2Mtt, when refer- 
ring to angles, as abbreviations of 90 , 180 , 360 and n times 360 
respectively. 

* More accurately, 

1 radian = 57.295, 779,5!3 degrees, 
1 degree = 0.017,453,290 radians, 
1 minute = 0.000,290,888 radians, 
1 second = 0.000,004,848 radians. 

The value of the radian in degrees has been calculated to 43 places of decimals. 



1 84 PLANE TRIGONOMETRY [chap, ix 

Exercise 44 

(Use 37- for ir and 57.3 for the value of 1 radian, n represents any 
positive integer.) 

1. Express decimally the following angles: 

45° 48' 36/' 185° 59 r 15", 35° 30' 30", 375° 00' 47". 

Ans. 45- 8l °; 185.9875°, 35-5°83°> 375- OI 3°5°- 

2. Express in degrees, minutes and seconds, 16.35°, I S3- I S^°t 
67.003°, I of a right angle. 

Ans. i6°2i', i53°09 / 2i.6 ,/ , 67° 00' 10.8", 64° 17' 08.57". 

3. Express in radians, 90°, 45°, 22J , 60°, 15°, 30°, 135°, 270°. 

o Oz-o A IT IT IT IT IT , 

-315 > -75 >36o . Ans.-, -, -, -, — , etc. 

2 4 8 3 12 

4. Express in sexagesimal units the following angles : 

IT IT 7T i 2 7T -, I 

— J J -> 22 7T, ■ J 2 7T, 2 W7T, 2, f , - ■ 

18 10 5 3 TT 

Ans. io°, i8°, . . . 114.6°, 28.65°, 18.23°. 

5. Give a geometrical representation of each of the following 
angles: 340°, f right angles, — f ir, 725°, 2 7r, 2 w 7r, (2 » + 1) w. 

6. State in which quadrant a line is after describing the following 

angles: 105°, -105°, ^, - |t, — , 2, +375°, 2W7r-j7r. 

4 3 

^4t?s. 2d, 3d, . . . 1st, 4th. 

7. Name three angles coterminal with an angle 30°. 

8. Name two positive and two negative angles coterminal with 
-i5°- 'Ans. 345°, 7°5°, "375°, ~735°- 

9. What is the smallest positive angle coterminal with 465 ? 
With -465°. Ans. 105°, 255°. 

10. Name the smallest negative angle coterminal with 735°? 
With -625 ? 

11. The principal value of an angle is | x; write in one formula all 
coterminal angles. Ans. 2 nir + \ ir. 

12. If 6 represents any angle less than 90°, all angles in the first 
quadrant can be expressed by 2 nir + 6. Express similarly all angles 
in each of the other three quadrants. 

Ans. II. (2n+ i)ir- 6. III. (2 n+ i)tt + d. IV. 2 mr — 6. 

13. What is the supplement of the angle \-k + 6? What is the 
complement? Ans. Jx — d, — 6. 



go] THE GENERAL ANGLE AND ITS MEASURES 185 

14. What is the complement of J tt + ? Of - - 0? 

3 

Ans. iir- 0,- + 0. 

15. Express both in radians and in degrees: 

(a) Each of the angles of an isosceles right triangle. 

(b) Each angle of a triangle whose angles are to each other as 

1:2:3. 

16. Find the number of radians in each of the angles of a regular 

, . . 3 7T ^ 7T (n— 2) T 

pentagon, octagon, n-gon. Ans. i2 — > - — > — 

S 4 n 

17. Express decimally the following angles : ij g 18' 19^, 25^65" 75"% 
187^05' 95^, i g oo v 15". Ans. 17. iSiq' 7 , 25.6575^, etc. 

18. Express 35 30' 30" and 35^30^30" each as a fraction of a right 
angle. Ans. 0.394537, 0.35303. 

19. Express 35 30' 30" in centesimal units. Ans. 39^ 45' 37^. 

20. Prove the following rule: To convert grades into degrees, 
diminish the number of grades by one- tenth of itself. 

21. Make a corresponding rule for converting degrees into grades. 

22. Find two regular polygons such that the number of degrees in 
an angle of one is to the number of degrees in an angle of the other 
as the number of sides of the first is to the number of sides of the 
second. Ans. Triangle and hexagon. 

23. Find three pairs of regular polygons such that the number of 
degrees in an angle of one is equal to the number of grades in an 
angle of the other. 

24. If g, d and r represent respectively the number of grades, 
degrees and radians in any angle, prove that 

n (g — d) = 20 r. 

90. Relation Between Angle, Arc and Radius. If an angle is 
subtended by an arc 20 ft. long and the radius of the circumference 
of which the arc forms a part is 10 ft., the number of radians in is 
20 ft. -t-'io ft. or 2, and generally, if an angle is subtended by an 
arc ^ units long and the radius measures r of the same units, the 
radian measure of the angle is 

e = 2. (1) 

r 



1 86 PLANE TRIGONOMETRY [chap, ix 

Solving this equation for s and r respectively, we get 

s = r6, (2) r= |. (3) 

Equation (1) enables us to find the angle when we know the length 
of the arc and the radius, (2) gives us the length of the arc in terms of 
the radius and the angle, and (3) gives the radius when the length of 
the arc is known as well as the angle which it subtends. 

These three equations enable us to solve a great variety of inter- 
esting problems. 

Example i. The diameter of the earth subtends an angle of 
17.5" at the center of the sun. Assuming the diameter of the earth 
to be 7917.6 miles, what is the distance of the earth from the sun? 

Solution. We may consider the diameter of the earth approxi- 
mately equal to an arc which subtends an angle of 17.5". Reducing 
17.5" to radians we find 

60 X 60 180 

hence, applying equation (1) we have for the required distance in 
miles 

5 7017.6 X 60 X 60 X 180 .1 

r — - — J -^— L = 93,322,000 miles. 

6 17.5 Xx 

(By the use of logarithms, putting ir = 3.1416.) 

Example 2. A railroad train going due north strikes a curve of 
3500 feet radius. The curve turns to the left and the train leaves 
the curve just 40 seconds after it entered the curve. In what direc- 
tion does the train move after it leaves the curve, assuming that it 
n is going at a uniform rate of 50 miles per hour. 

Solution. Let AN be the direction of the 
train on entering the curve and TB its direction 
when it leaves the curve at B. 

It is required to find the angle NTB or 0. 
^ Angle NTB = angle AOB (Why?), and by 
equation (1) 

arc AB 




Fig. 124. 



6 = angle AOB (in radians) = 



OA 



goa] THE GENERAL ANGLE AND ITS MEASURES 1 87 

OA is given and arc AB can be found, since we know the time which 
it takes the train to move over it at a given speed. 

OA = ^^ miles, arc AB = 5 ° X 4 ° miles, 

5280 60 X 60 

hence 

9= 50x40 ^asoo = ^ radianS) 

60 X 60 5280 105 



X 



180 88X180X7 _ 00 

— — 48 . 



IO5 7T IO5 X 22 

The train leaves the curve in the direction N. 48 W. 

90a. Area of a Circular Sector. The area of a circular sector is 
equal to the area of a triangle with a base equal in length to the 
arc of the sector and an altitude equal to the radius, hence the area 
A is 

A -2, 

2 

which by (2), Article 90, becomes 

A = \ r 3 9, 
where 6 is expressed in radians. (1) 

Example i . Find the area of a circular sector bounded by an arc 
10 feet in length, the radius of the circle being 50 feet. 
Solution. By (1), Article 90, the radian measure of the angle 

6 = — > and r = 50, hence by (1) 

A = - • S° ' — ' = 2 5° S Q- ft. 
2 50 

Exercise 45 

(To obtain the answers as given, use t = 37). 

1. Express in radians and in degrees the angle subtended by an 
arc 50 ft. long on a circle whose radius is 200 ft. 

Ans. \ radian = 14.3 . 

2. What is the radius of a circle on which an arc 100 ft. long sub- 
tends an angle of i°? Ans. — l — ft. 

7T 

3. How long is the arc subtended by a central angle of 75 , on a 
circumference whose radius is 100 ft.? Ans. 130.95 ft. 



1 88 PLANE TRIGONOMETRY [chap, ix 

4. Find the length of an arc of i° on the earth's surface (r = 3960 
miles). Ans. 69! miles. 

5. How long must a line be to subtend 1" at the distance of 
1 mile? Ans. 0.3 inch (approximately). 

6. A fly-wheel 10 ft. in diameter is revolving 200 revolutions per 
minute. Find the speed per second of a point on the rim. 

Ans. ^^ ft. 
3 

7. The diameter of the sun subtends an angle of 32' at the earth 
and the distance of the sun is about 93,000,000 miles. Required the 
diameter of the sun. Ans. 866,000 miles. 

8. The earth moves around the sun once each year. Assuming its 
path to be a circle, find the velocity of the earth per second. 

(Use the distance of the sun given in Problem 7.) 

Ans. 18.5 miles. 

9. The earth revolves on its axis once in 24 hours. Find its 
angular velocity per second, that is, the angle through which the earth 
turns per second, and hence the velocity in miles of a point on the 
equator. Use r = 3960 miles. 

Ans. Angular velocity 15" per sec, linear velocity 0.288 mi. per sec. 

10. The latitude of the city of Seattle is 47°4o'. Find the short- 
est distance from Seattle to the north pole. Ans. 2927 miles. 

(Distances on the earth are of course measured along the arc of a 
great circle. Use r = 3960 mi.) 

11. The diameter of a graduated circle is 12 inches and the gradu- 
ations on its rim are 15'. Find the approximate distance between 

two consecutive divisions on the rim. Ans. — of an inch. 



40 



12. Two. railroads meet at right angles at O. 
They are connected by a quadrant of a circle. 
The inner curve is 2000 ft. long. What is the 
distance of either point A or B from O ? 

Ans. OA=^ ft . 



7T 




13. In Problem 12, if the rails are 4 ft. apart, 
how much longer will be the outer rail than the inner rail of the 
curve? Ans. 2 ir ft. 



9i] THE GENERAL ANGLE AND ITS MEASURES 189 

14. A circular sector, radius 10 inches, has an area of 262 4 T sq. in. 
Find the angle of the sector. Ans. 30 . 

15. A city lot has the shape of a circular sector, the curve border- 
ing on the street. The straight sides of the lot are 100 ft. each and 
the angle between them is 6o°. The lot was sold at $100 per foot 
frontage, what was the price per acre ? Ans. $87,120. 

91. Review. 

1. (a) State and prove the law of sines, (b) State and prove the 

law of cosines, (c) Show that 2 R = — = = , where 

sin A sin B sin C 

R is the radius of the circumscribed circle, (d) Prove the projection 

theorem by means of the law of cosines. 

2. (a) In any triangle show that sin (A + B) = sin C, cos (A + B) 
= - cos C, sin i(A + B) = cos \ C, tan J (A + B) = cot \ C. (b) 
Prove the double formula and the law of tangents, (c) Apply the law 
of tangents to the two legs a and b and the opposite acute angles of 

a right triangle and obtain tan \ (A — B) = - • 

a + b 

3 . Express the area of a triangle : (a) In terms of two sides and the 
included angle, (b) In terms of the three sides, (c) Interpret geometri- 
cally the quantities s — a,s — b,s — c, and k = 1 / — — -• 

4. Show how to solve a triangle when the three sides are given, — 
(a) Without logarithms, (b) With logarithms, (c) Write down 

the formulas needed in (a) and (b). (d) Prove the half angle 
formulas in terms of the sides. 

5. Discuss the solution of each of the four cases of oblique tri- 
angles giving in each case the formulas necessary for the logarithmic 
solution and a formula for checking the answers. 

6. Review the method of solving each of the four cases of oblique 
triangles by division into right triangles. 

7. Given the three sides of a triangle as follows: 

a = 301.9, b = 673.1, c = 422.8, 

Compute one or more of each of the following sets of related quan- 
tities: 



190 PLANE TRIGONOMETRY [chap.dc 

(a) Angles, A = 18 12.4', B = 135 50.8', C = 2 5 56. 9 '. 

(b) Altitudes, ^=294.5, ^=132.1, ^=210.3. 

(c) Medians, 7^ = 541.4, #^=147.3, ^=476.9. 

(d) Angle bisectors, £4 = 512.8, 65=132.4, bc= 406.2. 

(e) Area and radii of 
inscribed and cir- 
cumscribed circles, T =44,458, r =63.61,, R =483.1. 

(/) Radii of escribed 

circles, r a =ii2.o, ^=1723, r c =i6i.o. 

(g) Check results by the graphic method. 

8. (a) Give a general definition of an angle of any magnitude. 

(b) Define a negative angle, (c) Give a general definition of the 
complement and supplement of an angle, (d) What is meant by 
the principal value of a set of coterminal angles, (e) Give three 
positive and three negative angles coterminal with 30 . 

9. (a) Explain each system of angular measures and define the 
unit in each, (b) State some of the advantages of each system. 

(c) State the relation between radians and degrees, (d) Express 
the following relations in radian measure, sin (180 — 6) = sin d, 
tan (90 - 0) = cot d, cos 180 = - 1. 

10. Prove the following relations, where a denotes the side of any 
regular polygon, p the perimeter, n the number of sides, r the radius 
of the inscribed circle, R the radius of the circumscribed circle, and 
A the area of the polygon. 

(a) a = 2 R sin - = 2 r tan - • 

n n 

(b) p — 2 nR sin - = 2 nr tan - • 

n n 

/ \ a t>9 • 7T tt nar 9 , t 

{c) A = nR 2 sin - cos - = = nr 2 tan - . 

n n 2 n 



CHAPTER X 



FUNCTIONS OF ANY ANGLE 



In this chapter a right angle will be denoted by R, rather than by 
90 or 7r/2. The advantage of this notation is that it frees our results 
from any particular system of measure, that is, our results hold equally 
well whether the angles are expressed in degrees, grades or radians. 
Whenever R is expressed in a given unit it is understood that the 
other angles are expressed in the same unit. Thus, if we write 
sin (90 — 6) it is understood that 6 is expressed in degrees, but if 
we write sin (x/2 — 6), 6 is to be expressed in radians, while the expres- 
sion sin (R — 6) does riot specify the unit in which the angles are 
measured. 

92. Definition of the Trigonometric Functions of Any Angle. 

The functions of any angle, positive or negative, are defined in 
exactly the same way as were the functions of an angle less than 
180 . Let 6 be any angle. Take the vertex O of the angle for an 




Fig. 126. 



Fig. 127, 



Fig. i2< 



origin and the initial side of the angle for the positive direction of 
the x-axis. Let P represent any point (not the origin) on the ter- 
minal side of the angle. Let x and y denote the rectangular coordi- 
nates of P, and r its distance from the origin. Then whether 6 is 
positive or negative, and whether P falls in the first, second, third 
or fourth quadrant, we have in every case, 



sifl e = v = ordinate, 
r distance 



esc 9 = — — -> 
sm0 



iqx 



192 



PLANE TRIGONOMETRY 



[chap. X 



_ A oc abscissa 
cos = — = — > 

r distance 

tane = g = ordinate . 
oc abscissa 



sec 8 = ■ 
cot6 = 



cos 

1 



tan6 



93. Signs of the Functions in Each of the Quadrants. 

(a) Since the sine is the ratio of the ordinate of P to its distance 
from 0, and the distance is always positive, the sign of the sine is 
the same as the sign of the ordinate, which is + in the first and 
second, — in the third and fourth quadrants. 

(b) Since the cosine is the ratio of the abscissa of P to its distance 
from 0, and the distance is always positive, the sign of the cosine is 
the same as the sign of the abscissa, which is + in the first and 
fourth, — in the second and third quadrants. 

(c) Since the tangent is the ratio of the ordinate of P to the 
abscissa of P, the sign of the tangent is + when the ordinate and 
abscissa have like signs, that is, in the first and third quadrants, 
and — when they have unlike signs, that is, in the second and fourth 
quadrants. 

(d) Any number and its reciprocal have like signs, hence the signs 
of the cosecant, secant and cotangent are the same as the signs of 
the sine, cosine and tangent respectively. 

The student must make himself perfectly familiar with the signs 
of the functions in the various quadrants. The following figure will 
prove an aid to his memory. 



tan 




94. Periodicity of the Trigonometric Functions. If the- ter- 
minal side of an angle is revolved in the plane of the angle through 
4 R, or any number of times 4 R, it will return to the position from 
which it started, and this is true whether the revolution is in the 
positive (counterclockwise) or negative (clockwise) direction. It fol- 
lows that the trigonometric functions of any angle remain unchanged 



95] 



FUNCTIONS OF ANY ANGLE 



J 93 



when the angle is increased or diminished by 4 R or by any number 
of times 4 R. That is, 

sin (0 ± 4 R) = sin 6, 

and similarly for each of the other functions. In general, 

sin (0 ± 4 nR) = sin 0, 

cos (8± 4 nH) = cos 0, 

tan (0 ± 4 nJR) = tan 0, etc., 
where n is any positive or negative integer. 



(1) 



Thus, 



sin 375° = sin (375° ~ 3 6o °) = sin 15 , 
sin (- 15 ) = sin (- 15 + 360 ) = sin345 c 



sin 



. 07T • /Q7T 

sin — = sin ' 
4 

7 TV 



2 7T = sin 



TV 



= sin 



77T 



+ 27T 



= cm ^ — , 



sm 



and similarly for any other function. 

It should be observed that by means of formula (1) the function 
of any negative angle can be replaced by the same function of some 
positive angle. 

Since the trigonometric functions remain unchanged when the 
angle is increased or diminished by 4 R, they are called periodic 
functions with the period 4 R. It will be shown presently that the 
tangent and cotangent have the smaller period 2 R. There are 
other periodic functions besides the trigonometric functions. 

95. Changes in the Values of the Functions while the Angle 
Changes from to 4 R. Let a point P start from a position A and 
move in the positive direction along the circum- 
ference of a circle whose radius OA equals unity. 
Join P to the center of the circle and let x and 
a y represent the coordinates of P in its various 
positions with reference to O as origin and OA as 
the positive x-axis. Let us consider the changes 
in the values of the various functions of the 
g * I 3 1 - angle A OP = 0, as the point P moves along the 

circumference of the circle. 




194 PLANE TRIGONOMETRY [chap, x 

By definition sin 6 = -2-, cos = — , tan d = %• But OP was 

OP OP x 

taken equal to i, hence sin 6 is represented by the ordinate y, cos 6 

by the abscissa x, and tan 6 by the ratio of the two. 

First quadrant. While P moves from A to B, that is, 
while 6 changes from o to R, 
y is positive and changes from otoi, 
hence sin 6 is positive and changes from o to i ; 

x is positive and changes from i to o, 
hence cos 6 is positive and changes from i to o; 

y/x is positive and changes from o to oo , 
hence tan 6 is positive and changes from o to oo . 

Second quadrant. While P moves from B to A f , that is 
while 6 changes from R to 2 R, 
y is positive and changes from 1 to o, 

hence sin 6 is positive and changes from 1 to o; 

x is negative and changes from — o to — 1, 

hence cos 6 is negative and changes from — o to — 1 ; 

y/x is negative and changes from — 00 to — o, 
hence tan 6 is negative and changes from — 00 to — o.- 

Third quadrant. While P moves from A' to B\ that is, 
while 6 changes from 2 R to 3 R, 
y is negative and changes from — o to — 1, 
hence sin 6 is negative and changes from — o to — 1 ; 

x is negative and changes from — 1 to — o, 
hence cos 6 is negative and changes from — 1 to — o; 

y/x is positive and changes from o to 00 , 
hence tan 6 is positive and changes from o to 00 . 

Fourth quadrant. While P moves from B' to A, that is, 
while 6 changes from 3 R to 4 R, 
y is negative and changes from — 1 to — o, 
hence sin 6 is negative and changes from — 1 to — o; 

x is positive and changes from o to 1, 
hence cos 6 is positive and changes from o to 1 ; 

y/x is negative and changes from — oo to — o, 
hence tan 6 is negative and changes from — 00 to — o. 




97] FUNCTIONS OF ANY ANGLE 1 95 

Cosecant, secant and cotangent. Since these functions are the 
reciprocals of the sine, cosine and tangent respectively, their varia- 
tions can be immediately written down from the variations of the 
latter. Remember, — 

(a) That the reciprocal of a number less than 1 is some number 
greater than 1, and vice versa. 

(b) That the reciprocal of o is 00 , and vice versa. 

(c) That reciprocals have like signs. 

96. Changes in the Value of the Tangent of an Angle as the 
Angle Changes From to 4 R. Some students rind it difficult to 
follow the changes in the tangent from the ratio 
y/x when x and y both change. The following dis- 
cussion is free from this difficulty. 

Let P (Fig. 132) move as in Fig. 131, but instead 
of the coordinates of the point P let us consider the 
coordinates of the point T or T' in which OP pro- 
duced meets one of the tangents to the circle at 

First quadrant. While 6 changes from o to R, AT changes from 

AT AT 
o to 00 , therefore tan 6 = = changes from o to 00 . 

OA 1 5 

Second quadrant. While 6 changes from R to 2 R, A'T' changes 

A'T' A'T' 

from + 00 to -J- o> therefore tan 6 = — — - = changes from — 00 

OA — 1 

to — o. 

Third quadrant. While 6 changes from 2 R to 3 R, A'T' changes 

A'T' A'T' 

from — o to — 00 , therefore tan 6 = = changes from + o 

OA' - 1 6 

tO + 00. 

Fourth quadrant. While 6 changes from 3 R to 4 R, AT changes 

AT AT 

from — 00 to — o, therefore tan 6 = — = changes from — 00 

OA 1 6 

to — o. 



97. Summary of Results. The results of the two preceding 

articles are brought together in the following table: 



196 



PLANE TRIGONOMETRY 



[chap. X 



Quadrant. 


I. 


II. 


III. 


IV. 


Angle 


to R 


2? to 2 R 


2 R to 3 R 


3 R to 4 R 


sin 


+ to + 1 


+ 1 to + 


— to — I 


— 1 to — 


cos 


+ 1 to -f 


— to — I 


— 1 to — 


+ to + I 


tan 


+ to + 00 


— 00 to — 


+ to + 00 


— 00 to — 


CSC 


+ 00 to + I 


+ 1 to + 00 


— 00 to — I 


— 1 to — 00 


sec 


+ 1 to + 00 


— GO to — I 


— 1 to — 00 


+ 00 to + I 


cot 


+ 00 to + 


— to — 00 


+ 00 to + 


— to — 00 



The student should observe, — 

(a) Every sine and cosine has some value between + 1 and — 1. 

(b) Every secant and cosecant has some value either greater 
than + 1, or less than — 1. 

(c) A tangent or cotangent may have any value whatever. 

(d) The functions change only at the points between the quad- 
rants, that is, when the angle has one of the values R, 2 R, 3 R, 4 R, 
etc, and then only when the value of the function is either o or 00 . 

(e) To a given value of a function correspond in general two differ- 
ent angles between o and 4 R. To a positive sine correspond two 
angles, one in the first the other in the second quadrant; to a nega- 
tive sine correspond two angles, one in the third the other in the 
fourth quadrant. To a negative tangent correspond two angles, one 
in the second the other in the fourth quadrant, etc. 



98. Fundamental Relations. All the fundamental relations be- 
tween the functions of an acute angle (Article 12) hold true when 
the angle is unrestricted in magnitude. The argument is an exact 
repetition of that used in Article 56. It follows that all trigonometric 
identities which have been proven for the case when the angle does 
not exceed a right angle, hold universally, that is, whatever be the 
magnitude of the angle provided that radical expressions such as 
Vi — cos 2 0, V 1 + tan 2 0, etc., be given the proper sign, + or — , 
depending on the quadrant in which 6 lies. 

99. Representation of Trigonometric Functions by Lines. 

Until recent times the trigonometric functions were defined by lines 
connected with a circle as follows: 

Let AOP be any angle, AP the arc which this angle intercepts on 
a circle drawn with as a center and any length OA as a radius. 



99] 



FUNCTIONS OF ANY ANGLE 



197 



Draw the radius OA ' perpendicular to OA, the initial side of the 
angle. Draw tangents to the circle at A and A', and produce OP to 
intersect these tangents in T and T' respectively. From P draw the 
perpendiculars PF and PF' to OA and OA' (produced if necessary) 
respectively. 





Fig- 134- 

The former definitions were then as follows: 

FP = sine of arc A P. 
F'P = sine of complementary arc A'P = cosine of arc AP. 

AT — tangent of arc AP. 
A'T' = tangent of complementary arc A'P = cotangent of arc AP. 

OT = secant of arc AP. 
OT' = secant of complementary arc A'P = cosecant of arc AP. 

FA = versine of arc AP. 
F'A' = versine of complementary arc A'P = co versine of arc AP. 

The definitions just given apply to any arc, provided the conven- 
tions regarding the algebraic signs of the various lines be carefully 
observed. These conventions are, as already stated in Article 53, 
with the additional one that the distances OT, OT' are positive if 
they pass through the extremity of the arc in question, that is, if the 
point P lies between and T or T' ; and negative if they do not, that 
is, if the point O lies between P and T or T'. Thus for an arc AP in 
the second quadrant, Fig. 135, 



FP, the sine, is positive; 
OF, the cosine, is negative; 
AT, the tangent, is negative; 
A'T', the cotangent, is negative; 
OT, the secant, is negative; 
OT', the cosecant, is positive. 

According to the old definitions, the length of each of the lines 
which defines the function depends on the length of the arc; and 




198 PLANE TRIGONOMETRY [chap, x 

since the length of the arc depends on the radius, it was necessary to 
specify the length of the radius employed. If, however, all lengths 
are expressed in terms of the radius as unit, the old definitions agree 
with the modern definitions. Thus in Fig. 133, 

tan arc A P = AT (old definition). 

arc AP 

Now = 6, the measure of the angle which the arc subtends, 

OA 

AT 

and = the measure of A T using OA as the unit of measure, 

OA 

so that, if we substitute for the actual lengths of the arc AP and the 
line AT their measures in terms of the radius, the old definition 

becomes 

AT 
tan d = — , which is the modern definition. 

OA 9 

Similarly, the old and the new definitions of each of the other 
functions may be shown to agree. 

The definitions of the functions as lines, while no longer used as 
definitions, are still useful in many ways. By their means the vari- 
ation of the functions in the various quadrants is most readily traced, 
and the fundamental relations sin 2 6 + cos 2 = 1, tan 2 6 + 1 = sec 2 0, 
cot 2 6 + 1 = esc 2 6 become manifest at sight. Above all, they ex- 
plain the origin of the names of the functions.* 

Exercise 46 

1. Make out a table giving each of the functions of 45 , 135 , 

o o 

225 , 315 . 

2. Make out a table giving each of the functions of 30 , 150 , 210 , 

o 

33° • 

* In the light of the historic definitions the origin of the terms tangent and 
secant is obvious. The origin of the terms cosine, cotangent and cosecant has 
already been explained. The origin of the term sine is probably as follows. The 
Latin word from which the word sine is derived is sinus, meaning " bay " or 
''bosom." The Arabic word was dschiba, meaning " half the chord of double an 
arc." Owing to the practice of the Arabs to omit the vowels in writing, dschiba 
was confused with dschaib meaning " bay " or " bosom," and it was this word 
dschaib which the Roman translators properly rendered sinus. The word arc 
comes from the Latin arcus, meaning "a bow." The versed sine was formerly 
called sagitta, an arrow, because it occupied the position of the arrow in a bow. 

The modern conception of the functions as ratios dates from the second half 
of the seventeenth century. The old definitions modified by using unity for the 
radius of the circle were used by many writers less than twenty-five years ago. 



ioi] FUNCTIONS OF ANY ANGLE 1 99 

3. Given sin 6 = §, find the values of 6 less than 4 R. What will 
be the values less than 4 R which 6 may have if cos 6 = \ ? 

Ans. 30 , 150 ; 6o°, 300 . 

4. Find the values of the following functions: sin 390 , cos 765 , 
tan 405 , sin (- 45 ), cos (- 30 ), tan (- 135 ): 

Ans. J, \ V 2 , 1- i V 2 , J V 3 , 1. 

5. Trace the changes of the cosine through each of the four quad- 
rants from the changes of the sine by means of the relation 

cos# = Vi — sin 2 6. 

6. Trace the changes of the secant in the first quadrant from 
those of the tangent by means of the relation sec 6 = V 1 + tan 2 6. 

7. Construct the lines representing the various functions of an 
arc in the third quadrant; of an arc in the fourth quadrant. 



8. What sign must be attached to the radical in sin 6 = v 1 — cos 2 6, 
when 6 is an angle in the second quadrant ? In the third quadrant ? 

9. What sign must be attached to the radical in 

sectf = Vi + tan 2 0, 
when 6 is an angle in the third quadrant ? In the fourth quadrant ? 

100. Reduction of Trigonometric Functions to the First 
Quadrant. In Articles 57 and 58 it was shown how to express any 
function of an angle in the second quadrant in terms of functions 
of an angle less than R. It remains to be shown how functions of 
angles in the third and fourth quadrants may be expressed in terms 
of functions of angles less than R. When this has been done, the 
value of any function of any angle can be found from the tables 
which contain the functions of angles in the first quadrant, that is, 
from o° to 90 . 

101. Reductions from the Third Quadrant. Any angle 6 3 in 
the third quadrant lies between 2 R and 3 R, hence every such angle 
may be expressed by either 

3 = 2 R + 6, or d 3 = 3 R - (f>, 
where 6 and 4> is eacn l ess than R. 



200 



PLANE TRIGONOMETRY 



[chap. X 



(a) 3 = 2 R + 0. Let 3 = angle AOP3 represent any angle in 
the third quadrant, and let = angle A'OP$. 
Produce P£) to P, making OP = OP 3 = r, then 
angle A OP = 0. Let (x, y),' (#3, ^3) denote the 
coordinates of P and P& respectively. Draw PF 
and P3F3 perpendicular to A A', then the triangles 
OPF and OP 3 F s are geometrically equal and 
yz = — v, #3 = — x. Hence we have, — 




Fig. 136. 



sin 6 3 = sin (2 M + 6) = ^ - — ? 

COS0 3 =COS(2JJ+e)= -= — 



= - - =- sine, 



.V 



tane 3 =tan(2JR + 0) = ^ 3 = — I 

X3 x 

From (^4) we obtain 
esc 3 = csc(2i?+ 0) = 

sec 3 = sec (2 R-\- d) = 

cot 3 = cot (2 7? + 0) = 



= = — cos 6, 

r 

1 = tane. 

x 



{A) 



sin (2 7? + 0) 



sin0 
1 



= — CSC 



cos (2 22+ 0) — cos 



= — sec d, 



= cot 6. 



(A') 



tan (2 7? + 0) tan0 

Observing that in both (^4) and {A') the signs on the right are 
the signs of the functions in the third quadrant, we have the simple 
rule: 

Any function of (2 R -\- 8) is equal to plus or minus the same function 
of 0, the sign being that of the function in the third quadrant. 

Example, sin 204 = sin (180 + 24°) = — sin 24 = — 0.4067, 
cos 204 = cos (180 + 24 ) = — cos 24 = — 0.9135, 
tan 204 = tan (180 + 24 ) = tan 24 = 0.4453. 

(b) 3 = 3#-<£- 

Put R — (f> = 0, then sin = cos 0, cos = sin </>, tan = cot <j). 
Hence v 

sin 3 = sin (3 It — <|>) = sin (2 R + 0) = — sin = — cos <|>, 

cos 3 = cos (3 R — <(>) = cos (2 R + 0) = — cos = — sin c|>, [ (B) 

tan 3 = tan (3 M — <|>) = tan (2 R + 0) = tan 0= cot ((>, 



io2] FUNCTIONS OF ANY ANGLE 201 



and from (B) 






esc 3 = esc (3R- 


6) - * 


1 j. ] 


sin (3 i? - 


J.\ ~ L ~ 0CC9, 

-(/)) — COS <j> 


sec 3 = sec (3 R - 


r^ - * 


1 J 


COS (3 A - 


-9) — sin <f> 


cot 3 = cot (3 R - 


c^ - X 


1 ^ 


tan (3 i? - 


— , — tan®. 
-</>) cot</> Y ) 



(B') 



In (5) and (B r ) the signs on the right are again the signs in the 
third quadrant of the functions on the left, hence the second rule: 

Any function of (j R — <j>) is equal to plus or minus the correspond- 
ing cofunction of (f>, the sign being that of the functions in the third 
quadrant. 

Example, sin 204 = sin (270 — 66°) = — cos 66° = — 0.4067, 
cos 204 = cos (270 — 66°) = — sin 66° = — 0.9135, 
tan 204 = tan (270 — 66) = cot 66 = 0.4453. 

102. Reductions from the Fourth Quadrant. Any angle 4 in 
the fourth quadrant lies between 3 R and 4 R, hence every such 
angle can be expressed by either 

6i = 4R-d, or 4 = 3^ + 0, 

where and <j> are angles less than R. 

(a) 4 = 4 R — 0. Let 4 = angle AOP± be any angle in the fourth 
quadrant, and let us put angle P4OA = 0. Draw 
OP = OP 4 = r, making angle AOP = 0. From P 
and P4 draw perpendiculars to OA. Then the A ' 
triangles OPF and OP±F are geometrically equal, 
and if (x, y), (x 4 , 3/4) denote the coordinates of P 
and Pa respectively, we have y± = — y, x± = x. Fig. 137. 

Consequently 



(A) 




sin 4 = sin (4 H - 0) = y - = — 2 = 

r r 


-2 = 
r 


1 

— sin0, 


cos 4 = cos (4 H — 0) = — = - 

r r 


= 


COS0, 


tan 04= tan (4^-0)=^ = ^ = 

X4 iA- 


~ 2 = 

X 


— tan0. 

J 



202 PLANE TRIGONOMETRY [chap, x 



From (A) follows 








esc 04 = esc (4 R - 


8) ~ ' I 


1 


- CSC0, 


sin (4 R - 0) 


— sin0 


sec 04 = sec (4 R - 


e) - J 


1 

COS0 


sec0, 


" J cos (412-0) 


cot 04 = cot (4 R - 


e) - T 


I 


- cot 0. 


tan (4 R - 0) 


— tan0 



{A') 



In (^4) and {A') the signs on the right are the signs of the functions 
in the fourth quadrant, hence 

Any function of (4 R — 0) is equal to plus or minus the same function 
of 0, the sign being that of the function in the fourth quadrant. 

Example, sin — - = sin [ 2t — - 1 = — sin- = - 



6/ 6 

T ^ - - - 1 = i \/ - 



COS ■ = COS 2 7T = COS - = f V 2, 

6 \ 6/ 6 2 ^ 

tan = tan f 2 w 1 = — tan - = — \ V 3. 



(5) 4 = 3# + <£. 

Put cf) = R — 6, then sin = cos 0, cos <£ = sin 0, cot <^> = tan 0, 
hence 



sin 9 4 = sin (3 It + <|>) = sin (4 R — 0) = — sin = — cos <t>, 

cos 4 = cos (3 It + <(>) = cos (4 R — 0) = cos = sin <|>, 

tan 84 = tan (3 JB + <|>) = tan (4 R— 0) = — tan = — cot <J>, 

and from (B) 

esc 4 = esc (3 R-\- <f>)= — — — = = — sec </>, 

sm (3 R + 9) — cos 9 

sec0 4 = sec (3 !? + <£) = ■ * = -7^— = csc0, 

cos (3 1c + 9) sin 9 

cot0 4 = cot (3 £ + <£) = T = L — - = -tan<£. 

tan (3 ic+ 9) — cot 9 



(B) 



(5') 



In (B) and (!>') the signs on the right are the signs in the fourth 
quadrant of the functions on the left, hence 

Any function of (3 R + 0) is equal to plus or minus the correspond- 
ing cofunction of <f>, the sign being that of the function in the fourth 
quadrant. 



103] 



FUNCTIONS OF ANY ANGLE 



203 



Example, sin - - = sin ( ^ + - ) = 



II 7T 

cos = cos 



(& + *)= 

\ 2 3/ 

tan ^-^ = tanf ^ + - )= — 
6 U 3/ 



7T 

cos- = 



. 7T 

sin- = 



1 



w 



2 V 3i 



cot- = 



-*v 3s 



103. Functions of Negative Angles. 

Let — 6 = angle ^40P' be any negative angle. 

Construct the angle ,4 OP = 0. Take OP = OP' = r, 

1 4 and let (x, y), (V, y') denote the coordinates of 

the points P and P' respectively, then x' — x, 

y' = — y, and we have 




Fig. 138. 



sin (- 0) = 



_3L - 



x 



X 



cos (- 8) = - = 
r r 

tan (-9)= * =' 

X 



X 



= — sin 6, 

= COS 0, } 

--tan0, 



(A) 



and from (A) 

esc (- 0) = 

sec (- B) = 

cot (- 6) = 





1 






I 


sin 


(- 


») 




— sin0 




1 






1 


cos 


(- 


■6) 




cos# 




1 






1 



tan (- d) 



= — CSC 



= sec 



tan# 



= — cot 



(A') 



The signs on the right are the signs of the functions in the fourth 
quadrant, hence 

Any function of (— 6) is equal to plus or minus the same function of 
6, the sign being that of the function in the fourth quadrant. 

Example, sin (— 13 25') = — sin 13 25' = — 0.2320, 
cos (- 13 25O = cos 13 25' = 0.9727, 
tan (- 13 25') = - tan 13 25' = - 0.2385. 



204 



PLANE TRIGONOMETRY 



[chap, x 



104. Table of Principal Reduction Formulas and General 
Rules. 

The principal results of the last three articles, together with the 
corresponding results for the first and second quadrants (Articles 
IO > 57? 58), are brought together in the following table for purposes 
of comparison and reference. 

Quadrant I 

sin (R — (f>) = cos cf>, 
cos (R — <j>) = sin (f>, 
tan (R — (j>) = cot </>. 

Quadrant II 

sin (R + <j>) = cos <fi, 
cos (R + </>) = — sin <f>, 
tan (2? + 0) = — cot ^. 

Quadrant III 



sin (2 R — 8) = sin 0, 
cos (2 R — 6) = — cos 0, 
tan (2 22 - 0) = - tan 0. 

sin (2R + O) = — sin 0, 
cos (2 R + 0) = — cos 0, 
tan(2i? + 0) = tan0. 



sin (3 R — (f>) = — cos cf>, 
cos ($R — (j>) = — sin <f>, 
tan (3 i? — <f>) = cot 0. 



Quadrant IV 



sin (4 i? — 0) = sin (— 0) = — sin 6, 
cos (4 R — 0) = cos (— 0) = cos 6, 
tan (4 R - 0) = tan (- 0) = - tan 0. 



sin (3 i? + 0) = — cos </), 
cos (3 i? + <f>) = sin (£, 
tan (3 R + 0) = — cot <^. 



We observe that each equation on the left involves a pair of same- 
named functions and the coefficients of R are even numbers, 2 or 4. 
On the right each equation involves a pair of conamed functions and 
the coefficients of R are the odd numbers 1 and 3. In either case 
the signs are the signs of the functions in the respective quadrants. 

By increasing the angles on the left by multiples of 4 R (which 
will not change the value of the functions), we obtain formulas for 
the functions of 

6R-6, 6R + 6, SR-d, SR-\-9, ., ., ., 2nR±6, 

and by increasing the angles on the right by multiples of 4 R, we 
obtain 
5^ + ^,5^-0,7^ + ^,7^-^ ., ., ., (2n+i)R±<f>. 



io 4 ] FUNCTIONS OF ANY ANGLE 205 

All the foregoing results are therefore included in the two form- 
ulas, 

Any function (2 nR ± 0) = ± same function 0, 

Any function (2 n -\- 1 R ± cf>) = i co function </>, 

the sign being the sign of the function on the left in the quadrant in 
which the angle falls when or <f> are acute angles.* 

Exercise 47 

1. Express in terms of same-named functions of angles less than R, 

sin 146 , cos 235 , tan 317 , sin 2 -, cos-*— , tan -^— • 

484 

-4?w. sin 34 , — cos 55 , — tan 43 , — sin-, — cos-, — tan-- 

484 

2. Express in terms of cof unctions of angles less than R, 

a. o o 0/ 2 IT • 2o X , 1 t-» 

tan 95 , sin 272 , cos 115 10, sec — , sin , cot 3^ R. 

3 3 

A 4. -O o • o / 7T TV . R 

Ans. — cot 5 , — cos 2 , — sin 25 10 , — esc-, — cos-, — tan— • 

662 

3. Express in terms of functions of positive angles less than 45 ° 
sin 143 15', cos 143 15', tan 243 io r 15", sec 284 30', cot 127 . 
Ans. sin 36 45', — cos 36 45', cot 26 49/ 45", esc 14 30', — tan 37° 

4. Use natural functions table to find 

sin iii° 30', cos 253°i2,' tan 134 , sin 3i7°i5', cos 97°35 / . 

Ans. 0,9304, — 0.2890, — 1.0355, ~~ 0-6788, — 0.1320. 

5. Find sin (- 150 ), cos 3564 , tan (- 5445°) 3 sin (- 27-V 

cos (- ioo°). Am _ ^ 0.8090, - 1, - \ V2, - 0.1736. 

6. If sin 6 = 0.5831, what values less than 4 R may 6 have ? 

7. If tan = — 4.3897, and sin 6 is known to be positive, find the 
value of 6. Ans. 6 = 102 50'. 

8. If cos 6 = sin 147°, show that one value of is 303 . 

9. If sin = cos 5 0, show that one value of is 15 , and another 

75°- 

10. Given sin <j> = — 0.4561; find tan <j>. Ans. tan <f> = 0.5125. 

* These rules hold not only for the sine, cosine and tangent, but for the cose- 
cant, secant and cotangent as well. The latter have been omitted from the sum- 
mary on account of their lesser importance. 



2 06 PLANE TRIGONOMETRY [chap, x 

105. Generalization of the Preceding Reduction Formulas. 

In the proof for the formulas for the functions of a negative angle 
(Article 103), was not restricted in magnitude. These formulas 
therefore hold true for any angle, but in the formulas for the functions 
of 2 R + 0, 3 R — <j> (Article 101), and of 4 R — 0, 3 R -f- <j) (Article 
102), and (f> were assumed to be angles between o and R. This 
restriction is not necessary and will now be removed. In other 
words, we will now show that the formulas of Articles 101 and 102 
hold for any value of and </>. The complete list of reduction form- 
ulas includes the formulas for the functions of 2 R — and of 
R ± <j>, we shall show that these formulas also hold for any value 
of the angles. 

For the sake of brevity the proofs will be confined to the first 
one of the formulas in each set. The proofs of the other formulas 
are left as exercises for the student. 

(a) Functions of (2 R + 0). The angles (2 R + 0) and differ 
by 2 R no matter how large is and whether is positive or negative. 
Consequently the points P 3 and P (Fig. 136) must always lie on a 
straight line through the origin. The coordinates of P 3 and P will 
therefore be numerically equal but opposite in sign. Hence for any 
value of 0, positive or negative, 

sin (2 R + 0) = 2- 3 = ^ = - sin0. (1) 

r r 

This establishes the first of the relations (^4), Article 101, for every 
value of 6. 

(b) Functions of (2 R — 6). If in (1) we put for 6, — 6, we obtain 

sin(2i£- 0) = - sin(- 0). 

But by (^4), Article 103, sin (— 0) =— sin for every value of 0, 

therefore 

sin(2#- 0) = sin0. (2) 

This establishes the first of the relations in Article 57 for every value 
of 0. 

(c) Functions of (4R — 0). The functions of an angle are not 

changed if the angle is increased or diminished by 4 R (Article 94), 

hence 

sin ( 4 R — 0) = sin (— 0), for every value of 0. 



105] FUNCTIONS OF ANY ANGLE 207 

But sin (— 0) = — sin0, by Article 103, 

therefore sin (4 R — 0) = — sin 0. (3) 

This establishes the first of the relations (^4), Article 102, for every 
value of the angle. 

{d) Functions of (R — (f>). Let (f> be any angle and let cj> f be the 
smallest positive angle co terminal with </>. Then </>' can be written in 
one of the forms 

0, 2 R — 6, 2 R + 0, 4 R — 0, where is positive and less than R, 
according as <f>' is an angle in the first, second, third or fourth 
quadrant. 

If <f>' = 2 R - d, then 

sin (R - </>') = sin (- R + 0) = - sin (R - 0) = - cos0 = cose//. 

If(//= 2R + 6,then 

sin (R - 00 = sin (- R - 6) = - sin (R + 0) = - cos 6 = cos <£'. 

If ft = 4 R - 0, then 

sin (R - (f> f ) = sin (- 3 #+ 0) = -sin (3 R - 0) = cos0 = cos^'. 

We see then that, whether <// is in the first, second, third or fourth 

quadrant, 

sin (R — <j>') = cos ft, 
and hence 

sin (R — <f>) = cos (4) 

is established for every value of <fi. 

(e) Functions of (R + 0), (3 i? — 0), (3 i? + 0). Let <£ be any 
angle and put <^ = R — 6. Then by applying the formulas whose 
generality has been already established, we find 

sin (R + </>) = sin (2 R — 0) = sin = cos (5) 

sin (3 R — (f>) = sin (2 i? + 0) = — sin = — cos cj), (6) 

sin (3 R + 0) = sin (4 R — 0) = — sin = — cos <f>. (7) 

Exercise 48 

Repeat the argument of the preceding article to show that for all 
values of and cf> respectively, — 

1. tan (2R ± 6) = ± tan0. 

2. tan (R ± (f>) = =F cot 0. 

3. tan (3 i? ± cf>) = ± cot 0. 



208 PLANE TRIGONOMETRY [chap.x 

4. Show that for every value of 0, 

sin (0 — R) —— cos 0, sin (0 — 2 R) = — sin 0, sin (0 — 3 i?) = cos 0. 
cos (6 — R) = sin 0, cos (0 — 2 i?) = — cos 0, cos (d — 3 R) = — sin 0. 
tan ((9- 22) =- cot 0, tan (0— 2 R) = tan 0, tan (0- 3R) =— cot0. 

5. Prove geometrically that cos (2? — 0) = sin 0, 

(o) When <£ is an angle in the second quadrant. 
(&) When (56 is an angle in the third quadrant. 
(c) When cj) is an angle in the fourth quadrant. 

6. General proofs of the reduction formulas for the angles 

(2/2-0), UR-0), (2R + 6) 

may be obtained from considerations of symmetry. Referred to 
the same origin and taking the initial line to coincide with the 
it-axis, then no matter how large 0, and whether positive or negative, 

(a) The terminal sides of and (2 R — 0) are symmetrically 

situated with respect to the v-axis. 

(b) The terminal sides of and (4 R — 0) are symmetrically situ- 

ated with respect to the x-axis. 

(c) The terminal sides of and (2 R + 0) are symmetrically 

situated with respect to the origin. 

It follows that the same functions of each pair of angles are numer- 
ically equal and that in 

(a) The sines have like and the cosines opposite signs. 

(b) The sines have opposite and the cosines like signs. 

(c) The sines and cosines each have opposite signs. 

In each case the sign of the tangent may be determined from the 

relation 

, n sin 
tan = 

cos 



CHAPTER XI 
FUNCTIONS OF TWO OR MORE ANGLES 

106. Addition Theorems for the Sine and Cosine. 

First Proof. If a, b, c denote the sides of any triangle and A, B, C 
the angles opposite these sides, we have from the law of sines 



(Article 62, (3)), 






c 


a = D sin A, 






j/\\\ 


b = D sin B, 








c = D sin C, 


(Article 63, 




<*- — - 6 cos A- - ->k-a cos 2?V| 


and by the projection theorem 

(4)), ' 


A 


c=D sin C B 
Fig. 139. 



c = a cos B -\- b cos ^4 . 

Substituting in this equation the above values for a, b and c } and 
dividing out the constant factor D, we get 

sin C = sin A cos B + cos .4 sin B. 

Now C = 180 — (A + 5), therefore sin C — sin {A + B), whence 
sin (A + B) = sin ^4. cos B + cos ^ sin B. (1) 

Again 

cos 2 (A+B) = 1- sin 2 (4 + 5) 

= 1 — (sin A cos 5 + cos A sin i?) 2 

= 1 — sin 2 ^4 cos 2 i? — 2 sin ^4 sin B cos A cos jB — cos 2 A sin 2 5 

= 1 — (1 — cos 2 A) cos 2 2? — 2 (...)— (1 — sin 2 ^4) sin 2 B 

= cos 2 ^4 cos 2 B — 2 sin A sin 5 cos ^4 cos B + sin 2 ^4 sin 2 B 

= (cos ^4 cos B — sin A sin 2?) 2 . 

Taking the square root of both sides, 

cos (A + -B) = cos ^1 cos -B — sin A sin -B. (2) 

Since the last expression was obtained by extracting a square root 
it may seem that the double sign, ±, should have been put before 
the right-hand member of (2), but on putting B = o, the minus 

2og 



2IO PLANE TRIGONOMETRY [chap, xi 

would yield the result cos A — — cos A , which shows that the minus 
sign cannot be used. 

Formulas (i) and (2) embody the so-called addition theorems for 
the sine and cosine respectively. In words, — 

The sine of the sum of two angles is equal to the sine of the first angle 
times the cosine of the second plus the cosine of the first angle times the 
sine of the second. 

The cosine of the sum of two angles is equal to the product of the 
cosines of the separate angles diminished by the product of their sines. 

It is plain that by means of these theorems the sine and cosine of 
the sum of two angles may be found if the sines and cosines of each 
of the separate angles are known. 

Example i. Given the functions of 45 and of 30 , to find the 
sine and cosine of 75 . 

Solution, sin 75°= sin (45°+ 30 ) = sin 45 cos 30°+ cos 45 sin 30 

cos 75°= cos (45°+ 30 ) = cos 45 cos 30 — sin 45 sin 30 
= W2 -iV3 - W~2 -I = i (V6 - V2). 

107. Generalization of the Addition Theorems. In the fore- 
going demonstration, A and B are angles of a triangle, their sum is 
therefore necessarily less than 2 R. This restriction may be removed, 
in other words, the addition theorems hold for angles of any mag- 
nitude and whether positive or negative. 

To prove this let A and B be two angles each less than R, so that 
their sum is less than 2 R, and let A 1 = A ± R, then 

sin^4i = ± cos A, cosAi = =F sin ^4, (Art. 104, and Ex. 48, 4) 
sin (ili + B) = sin {A ± R + B) = ± cos (A + B) 

— ± (cos A cos B — sin A sin B) 

= ± cos A cos B =F sin A sin B 

= sin ^4i cos 2? -f- cos^4i sin 5. (1) 

cos (Ai + B) = cos {A ± R + B) = q= sin {A + B) 

= =F (sin A cos B + cos A sin B) 

= T sin A cos B =F cos A sin B 

= cos A 1 cos B — sin A 1 sin B * (2) 

* If the student finds it difficult to follow the double signs, let him consider the 
two cases Ai = A + R, A\ = A — R, separately. 



108] FUNCTIONS OF TWO OR MORE ANGLES 211 

Equations (i) and (2) show that the addition theorems continue 
to hold if the angle A is increased or diminished by R, and the same 
reasoning applies to the angle B. By a repetition of the process 
just employed it is clear that the theorems will continue to hold true 
if Ai is replaced by A 2 , where A 2 = Ai ± R = A\ ± 2 R, and gen- 
erally that A may be replaced by 

A n = A ± nR, 

and B by 

B m = B ± mR, 

n and m being two arbitrary integers. 

But this proves the theorems for all values of the angles, for any 
positive or negative angle may be put in the form A ± nR, where n 
is some integer and A some angle less than R. 

108. Addition Theorems. Second Proof. The addition theorems 
may be proved without making use of the law of sines and the pro- 
jection theorem. 

In Fig. 140, let XOM = angle A, and MON = 
angle B, then XON = angle (A + B). 

On ON take any point P, and from P draw 
PT and PQ perpendicular to OX and OM 
respectively. cm 

From Q draw QR perpendicular to OX and pj g I4Q 

QS parallel to OX. 

The triangles QOR and QPS are similar (Why?), hence angle 
QPS = angled. Now 

*n(A + B)=^= TS + SP = ZQ + SP 
K J OP OP OP OP 

^?Q . QQ + sp . QE 

~ OQ ' OP QP ' OP 
= sin A cos B + cos A sin B. 

cos(A+B)=°T = OR - TR = °K-SQ 
OP OP OP OP 

= QR . OQ _ SQ m QP 
" OQ * OP QP ' OP 

= cos A cos B — sin A sin B. 




212 PLANE TRIGONOMETRY [chap, xi 

In the figure we have taken A -\- B less than a right angle, but the 
proof just given will hold for any angles, provided proper attention 
be given to the algebraic signs of the lines which enter the figure. 

109. Subtraction Theorems for the Sine and Cosine. Since 
the addition theorems have been shown to hold for negative as well 
as for positive angles, we may replace B by — B. The equations (i) 
and (2), Article 106, then become 

sin (A — B) = sin A cos (— B) + cos A sin (— B) 

cos (A — B) = cos A cos (— B) — sin A sin (— B), 
from which 

sin {A — B) = sin A cos B — cos A sin B, (i). 

cos (A — B)= cos A cos B + sin A sin B. (2) 

These formulas enable us to compute the sine and cosine of the 
difference of two angles if the sines and cosines of the separate angles 
are known. 

Example. Given the functions of 45 and 30 ; to find the sine 
and cosine of 15 . 

Solution. 

sin 15° = sin (45 — 30 ) = sin 45 cos 30 — cos 45 sin 30 

= iv^.jV3-jV2.J = Hv / 6-\ / 2'). 

cos 15° = cos (45 — 30 ) = cos 45 cos 30 + sin 45 sin 30 

= l^.iv / 3 + |v / 2.i = i(v / 6 + V2). 

Exercise 49 

1. Find the sine and cosine of i5°from the relation, i5° = 6o°— 45 . 

2. Given sin x = f , cos x = |, sin y = T 8 T , cos y = \% ; find 
sin (x + y) and cos (x + y). 

3. Find sin oo° and cos oo° from the relation oo° = 6o° + 30 . 

4. Find sin o° and cos o° from the relation o° = 30 — 30 . 

5. Apply the addition and subtraction theorems to find the fol- 
lowing: 

sin (90 — x), cos (oo° + x), sin (180 + x), cos (270 — x), 
sin (360 — y), cos (45 - y), sin (30 + y), cos (6o° — y). 



log] 



FUNCTIONS OF TWO OR MORE ANGLES 



213 



6. Show that 

sin (A + B) + sin (A — B) = 2 sin A cos B, 
sin (A + B) — sin (A — B) = 2 cos ^4 sin B, 
cos (^4 + B) -f- cos (^4 — B) = 2 cos ^4 cos B, 
cos (^4 + B) — cos 04 — B) — — 2 sin A sin 5. 

7. By putting B = A in Problem 6 show that 

sin 2 ^4 = 2 sin A cos ^4 , cos 2 ^4 = 2 cos 2 A — 1 = 1 — 2 sin 2 ^4 . 

8. Prove the subtraction theorems geo- 
metrically by means of Fig. 141. XOQ = 
angle A, MON = angle B, XON = angle 
(A — B). QR, QP, PT are perpendiculars 
to OX, OM, OX respectively. Angle SQP = 
angle A (Why?). 

Qo Show that the answers to Problem 17, 
Exercise 38, may be put in the forms 

a sin (3 cos a 

sin (a — j8) 




Fig. 141. 



x 



_ a sin a sin ft 
sin (a — 0) 



10. Show that the answer to Problem 19, Exercise 38, may be 
written 



h = 



a sin a sin a 



V sin (a: — a/) sin (a + a') 



11. Show that 



and hence that 



cos ^4 cos B 



tan ^4 + tan B _ sin Q4 + i?) 
tan ^4 — tan i? sin (^4 — 5) 



12. Show that sin in + 1) = sin nd cos + cos nd sin 0, 
and hence that 
similarly 



sin n -}- 1 = sin ^° cos i° + cos #° sin i ( 



cos n + 1 = cos w° cos i° — sin w° sin i°. 

Hence, if the sine and cosine of i° are known, those of 2 , 3 , etc., 
may be readily computed. 

13. Show that cos (A + }?r) + sin (A — Jr) = o, 
and sin (4 + J tt) + cos (A — J ?r) = V2 (sin .4+ cos ^4). 



PLANE TRIGONOMETRY 



[chap. XI 



214 

14. Show that 

cos ,4 sin (B - C) + cos B sin (C - A) + cosCsin (4 - 5) = o, 
sin A sin (5 - C) + sin £ sin (C - 4) + sin C sin (4 - B) = o. 

15. Two straight roads OA and 05 (Fig. 142) 
cross at an angle a. From O, their point of inter- 
section, a straight road is to be laid out to a 
point P, which is p miles from the first road and 
q miles from the second. Required the angle 
which OP will make with OA . 




i? - sin 
q sin (a — 0) 



Fig. 142. 

whence tan 6 = — *- — 

p cos a + q 



1:6. The area T of a triangle was computed from two sides, b and c, 
and the included angle A. Afterwards it was found that an error e 
had been made in measuring the angle A . Show that the corrected 
area is given by the formula 

V — T (cos e + sin e cot A). 

17. Two parallel forces p and q act on levers of lengths a and b 
respectively, which are inclined at an angle a at 
the common fulcrum O. What angle 6 must the 
forces make with the lever a in order that there 
may be equilibrium? 

(Suggestion. Equating moments about O we 
have 




ap sin 6 = bq sin (a + 6), 
tan 6 = b 1 sin g ■) 



sin 2 B = cos 2 B — cos 2 A, 
sin 2 5 = cos 2 B — sin 2 ^4 . 



from which 

ap — bq cos a 

18. Show that 

sin (A + 5) sin (A - B) = sin 2 A 
cos (4 + 5) cos (A- B) = cos 2 ^ 

19. x, y, z being any angles, show that 

sin (x + y -f- z) = sin # cos 3/ cos z -f- sin 3; cos z cos # + sin z cos # cos 3; 

— sin x sin y sin z. 

cos (# + y + z) = cos x cos y cos z — cos x sin 3/ sin z — cos y sin z sin x 

— cos z sin x sin 3;. 



no] FUNCTIONS OF TWO OR MORE ANGLES 21 5 

20. Show that 

sin (x + y — z) + sin (x — y + z) + sin (— x -f- y + z) = 

sin (x + y + z) + 4 sin # sin v sin z. 

cos (# + y — z) + cos (x — y + z) + cos (— # + y + z) = 

4 cos x cos y cos z — cos (x -f- y + 2). 

21 . By eliminating a, b, c from the equations (Art. 63), 

c = a cos B + b cos ^4 , 

a = b cos C + c cos 5, 

& = c cos A -\- a cos C, 
show that 

cos 2 ^4 + cos 2 B + cos 2 C + 2 cos ^4 cos 5 cos C = 1. 

Solve this equation for cos C, and obtain 

cos C = — cos A cos 2? ± sin ^4 sin 5. 

Remembering that C = 180 — (A + B), and disregarding the 
lower sign which is inadmissible (Why?), we find 

cos (A + B) = cos A cos B — sin A sin J5. 

This constitutes another proof of the addition theorem for the 
cosine. 

110. Tangent of the Sum and Difference of Two Angles. If 

we divide the sine of the sum of two angles by the cosine we obtain 
the tangent, thus 

(a 1 72\ _ sin (A + B) _ sin A cos B -j- cos ^4 sin B 
cos (^4 + B) cos ^4 cos 5 — sin A sin 2? 

On dividing both the numerator and denominator of the right- 
hand member by cos A cos B, we have 

sin A cos B . cos A sin B sin ^4 . sin B 

f a 1 -d\ cos ^4 cos B cos yl cos i? cos A cos i? 

tan (A-j-B) = 



cos ^4 cos 5 sin A sin 5' sin A sin 2? 

£ j 1 — 

cos A cos 5 cos A cos 5 cos ^4 cos B 

that is 

Ha(i + J). tan^t + tani? (i) 

I — tan A tan i? 

To obtain the tangent of the difference of two angles, we need 
only put in (1) for B, — B, thus 

tan (A - B) = tan A + tan ( ~ B) - , 

i — tan ^4 tan (— 2?) 



216 PLANE TRIGONOMETRY [chap, xi 

that is, 

+«w a t>\ tan A — tan B , n 

tan (A — B) = — ■ (2) 

1 + tan A tan B 

Of course we might have deduced (2) just as we deduced (1) that 
is, by dividing sin (A — B) by cos (A — B). 

111. Functions of Double an Angle, li B = A, the formulas 
for the sine, cosine and tangent of the sum of two angles become 

sin {A + A) = sin A cos A + cos A sin A, 

cos (A + A) = cos A cos A — sin A sin A, 

( A , A N tan ^4 + tan ^4 

tan (A + A) = — -, 

1 — tan A tan A 

that is, 

sin 2 A = 2 sin A cos ^. (1) 

cos 2 A = cos 3 ^1 — sin 3 A = 1 — 2 sin 3 ^ = 2 cos 2 A—i, (2) 

, . 2 tan ^4. z x 

tan 2^= ; — (3) 

1 - tan* ^ uy 

By means of these formulas the functions of twice an angle are 
easily computed, provided the functions of the single angle are 
known. 

112. Functions of Half an Angle. It is frequently necessary to 
express the functions of half an angle in terms of the functions of 
the whole angle. This is most easily accomplished by means of (2), 
Article 111. Since these formulas hold for any value of the angle, 
we may replace A by J A, thus 

cos (2 • J ^4) = 1 — 2 sin 2 \ A = 2 cos 2 \ A — 1, 

or cos ^4 = i — 2 sin 2 \ A ) 

21 a ' \ W 

§ =2 cos 2 f A — 1. ) 

If we solve the first of these equations for sin J A , we obtain 

• 1 a .. li— cos A f x 

sm±A = ±\/ > (2) 

V 2 

and the second solved for cos J A gives 



cos|^ = ±y/ I + cos ^ - (3) 



ii2] FUNCTIONS OF TWO OR MORE ANGLES 217 

Dividing (2) by (3) gives 



= ±y/t^H- (4) 

▼ 1 + cos A 



tanM 

" + COS A 



Exercise 50 

1. Given the tangents of 45 and 30 , compute the tangents of 
75°andi 5 . 

Ans. tan 75°= 2-± — -3, tan 15°= ^-^ 5. 

3-^3 3 + ^3 

2. Given tan A = §, tan 5 = \, find tan (4. + B) and tan (4. — B). 

3. Show that 

/ o 1 a \ 1 + tan A . , o ,1 \ 1 — tan 4. 

tan (45° + ^) = — ! — ; -, tan (45 - 4) = — — -• 

1 — tan A 1 + tan 4. 

4. Show that 

*. t a 1 t>\ cot 4 cot B — 1 , / . D \ cot 4 cot B + 1 

cot L4 + 2?) = j cot Li — B) = — — - • 

cot4+cot£ -cot4+cot£ 

5. Show that 

/ , - n tan x + tan y + fean 2 — tan x tan 3/ tan z 

tan (# +3/ + z) = ! JL — ■ z 

1 — tan x tan y — tan y tan z — tan z tan x 

6. Express sin 4 4, cos 4. A, tan 4 v4 in terms of the functions 
of 2 A. 

7. Given the functions of 30 ; find the sine, cosine and tangent 
of 6o°. 



8. Given the functions of 45 ; find the sine, cosine and tangent 

2 



of 22 10 



Ans. sin 22j° = J\/2 — \/2, cos22j° = J V2+V2, tan 22j° = v 2 — 1. 

9. Given the functions of 30 ; find the sine, cosine and tangent 
of 15°. = = 

Ans. sini5° = | ^2-^3, cos 15°= J V 2 + V3, tan 15°= 2- V3. 

10. Express sin 3 4. in terms of sin A. 

Ans. sin 3 A = 3 sin A — 4 sin 3 A 
(Suggestion. 3 ^4 = 2 4. + -4.) 

11. Express cos 3 A in terms of cos A. 

Ans. cos 3^ = 4 cos 3 4 — 3 cos A. 



2l8 



PLANE TRIGONOMETRY 



[chap. XI 



12. Express tan 3 A in terms of tan A. 

A , 3 tan A — tan 3 A 

Ans. tan 3 A = l2 • 

1 — 3 tan 2 A 

13. Find the sine and cosine of 18 . 



a • 00 V 5 — 1 
^4?w. smi8 = ■ — 3 > 

4 



cos 18 = 



v/ io + 2\A; _ 



(Suggestion. Let x — 18 , then 2 # + 3 # = 90 , 2 x = 90 — 3 x, 
sin 2 x = cos 3 #. 

Now express sin 2 x and cos 3 # e ach in ter ms of functions of x, 
and solve for sin x. Then cos x = V 1 — sin 2 #.) 



14. Show that if / stands for tan A, 



A 2 1 
sin 2 A = — > 



A 1 ~ t2 
COS 2 A = : j 



I + t 2 I + t 2 

15. Use Fig. 144 to prove that 
sin 2 A = 2 sin A cos ^4 , 

cos 2 ^4 = cos 2 A — sin 2 A 

16. Use Fig. 145 to prove that 
sin J A = Vj (1 — cos A), 



tan 2 A = 



2/ 



i-* 2 




cos J A = \/\ (1 + cos A). 




RS 



(Suggestion, sin \ A = — , ftS" = OR' + OS . - 2 OR • 05 cos 4 , 

2 r 

cosi^ = — , ZP 2 = 0£ 2 +OP 2 + 2 0£* OP cos 4,) 
2 r 

17.- If ^ + B + C = 180 , show that 

tan A + tan B + tan C = tan ^4 tan B tan C. 

2 tan h A 



18. tan ^4 = 



- — . Solve this equation for tan J A , and 



1- tan 2 J .4 
identify your result with (4), Art. 112. 

19. Show that the equation a tan x + b cot x = c, may be re- 
duced to the form (a — b) cos 2 x + c sin 2 x = a + &. 



H3] FUNCTIONS OF TWO OR MORE ANGLES 219 

20. A flagpole 50 ft. high stands on a tower 40 ft. high. At what 
distance from the foot of the tower will the flagpole and tower sub- 
tend equal angles? Ans. 120 ft. 

21. A tower is situated at a distance of a ft. from the banks of a 
river b ft. wide. At what height on the tower will the river subtend 
an angle of 30 ? 

Ans. I b V3 ± \ V3 b 2 — 4 a 2 — 4 ab. (Two solutions?) 

22. The dial of a town clock has a diameter of 2 r = 8 ft. and its 
center is h = 70 ft. above the ground. At what distance from the 
foot of the tower will the dial be most plainly visible? 

Ans. x = V(h + r){h — r) = 69.98 ft. 

(Suggestion. The angle subtended by the dial must be the largest 
possible.) 

23. The height h of an object AB was computed from the dis- 
tance J of a point O from the foot of the object 
and the angle 6 which AB subtended at this point. 
It was found that an error e had been made in ,se, 

measuring 6. Show that h must be corrected by o ^^ e d ^ 

an amount Fig. 146. 

_ d sin e 

cos (6 + e) cos 6 

113. Sums and Differences Transformed into Products. From 

sin (A + B) = sin A cos B + cos A sin B 
sin {A — B) = sin A cos B — cos A sin B 
cos (A + B) = cos A cos B — sin A sin B 
cos (A — B) = cos A cos B + sin A sin 5 

we obtain by addition and subtraction 

sin {A + B) + sin {A — B) = 2 sin ^4 cos 5 
sin (A -\- B) — sin 04 — B) = 2 cos ^4 sin B 
cos 04 + B) + cos 04 — B) = 2 cos ^4 cos B 
cos 04 + B) — cos (4 — B) = — 2 sin A sin 5. 




(l) 



2 20 PLANE TRIGONOMETRY [chap, xi 

Let us now put 

A + B = x and A — B = y, 

from which A = \ (x + y), B = J (x — y), 

so that the preceding formulas become 

sin as + sin y = 2 sin | (as + y) cos | (x — yY 

sin x — sin y = 2 cos | (a? + y) sin J (as — ?/) 

cos as + cos y = 2 cos \ (as + 2/) cos | (as — y) 

cos as — cos 2/=— 2sin|(o5+y) sin | (as — 7/). 

These formulas are frequently used in the further study of mathe- 
matics. One of their uses is that they enable us to replace sums or 
differences by products and thus help us to adapt formulas to com- 
putation by logarithms. 

Example i . Transform sin 2 + sin 4 + sin 6 into a product. 
Solution. By the first formula (1) 

sin 2 + sin 4 = 2 sin J (2 -f- 4 0) cos J (2 — 4 0) 

= 2 sin 3 cos 
and by (1), Art. in, 

sin 6 = 2 sin 3 cos 3 0. 
Adding 

sin 2 + sin 4 -f- sin 6 = 2 sin 3 (cos 3 0+ cos 0) . 

We next apply the third of formula (1) to the expression in paren- 
theses on the right 

cos 3 0+ cos 0=2 cos 2 cos 0, 

so that finally 

sin 2 + sin 4 + sin 6 = 4 sin 3 cos 2 cos 0. 

Example 2. If ,4 + 2? + C= 180 , show that 
sin 2 A + sin 2 5 + sin 2 C = 4 sin ^4 sin 5 sin C. 

Solution, sin 2 A + sin 2 5= 2 sin (^4 + I>) cos (A — B ) 

= 2 sin C cos (^4 — 5), 
and sin 2 C= 2 sin C cos C 

= — 2 sin C cos (A -\- B); 



H3] FUNCTIONS OF TWO OR MORE ANGLES 2 21 

hence 

sin 2 A + sin 2 B + sin 2 C= 2 sin C [cos (^4 — B) — cos (^4 + B)]; 

but by the fourth of the formulas (i) 
cos ( A — B) — cos (A-\- B) = 2 sin ^4 sin B, 

therefore finally 

sin 2 A + sin 2 B + sin 2 C = 4 sin ^4 sin 5 sin C. 

Exercise 51 

1. State in words the theorems embodied in the formulas (1), 
Article 113. 



2. Show that 



3. Show that 



sin (30 + y) -f- sin (30 — y) = cos y, 
sin (30 + y) — sin (30 — y) = V 3 sin y, 
cos (30 + y) + cos (30 — y) = V 3 cos y, 
cos (30 + y) — cos (30 — y) = — sin y. 

sin 75°+ sin 15°= \ V6, 
sin 75 — sin 15°= J V2, 
cos 75°+ cos 15°= J v 6, 



cos 75 —cos 15 



°= — * VI 



4. Express the following products as sums or differences of two 
functions, 

sin id° cos 5 , cos 20 sin io°, sin \ •# sin J 0, 

COS (J X — «0) COS (J 7T + 0). 

^4^5. \ (sin 1 5 + sin 5 ), .... J (cos J 7r + cos 2 0) = § cos 2 0. 

5. Show that 

sin i6°+ sin 14°= 2 sin 15 cos i°, 

. Z X . X . X 

sin a sm - = 2 cos x sin — , 

22 2 

sin (» + 1) x + sin (n — 1) x = 2 sin nx cos #. 



2 2 2 PLANE TRIGONOMETRY [chap, xi 

Prove the following identities: 

, sin A + sin B _ tan \ (A -f- B) 
sin A — sin B tan J (4 — B) 

COS ^4 + COS B t.1 f a \ r>\ 4. i / a m 

7- - — t- 1 ^ = - cot i (A + £) cot i\A — B). 

cos ^4 — cos B 

sin ^4 — sin B ■, , A d n 

= tan| {A — B). 



cos A — cos B 
sin .4 — sin B 



= - coti(A + B). 



cos A — • cos i> 

io. Show that 

sin (a + x) sin # = w cos (a + x) cos x 
may be transformed into 

— cos (a + 2 x) + cos a — m [cos (a + 2 x) + cos a]. 

1 1 . Show that 

\ (cos 2 x + cos 23/) = cos (x + 3;) cos (# — y), 

— J (cos 2 x — cos 23/)= sin (re + y) sin (x — 3/). 

12. Using the results of 11 show that 

cos (x-{-y) sin (x— y)-\-cos (y-\-z) sin (y— z)+cos (z+x) sin (z— x) = o. 

13. Show that 

sin ioo°+ sin 40°+ sin 6o°= 4 cos 30 ° sin 50 cos 20 , 
sin 2 d + sin 6 + sin 8 = 4 cos cos 3 sin 4 0. 

14. Show that 

sin x + sin (x — 2) = 2 sin (# — 1) cos 1, 

whence sin x = 2 sin (x — 1) cos 1 — sin (x — 2). 

Similarly sin x = 2 cos (# — 1) sin 1 + sin (x — 2), 

cos x = 2 cos (x — 1) cos 1 — cos (x — 2), 

cos x = — 2 sin (# — 1) sin 1 + cos (x — 2). 

These formulas enable us to compute the sine or cosine of x° if 
the sines and cosines of (x — i)°, (x — 2) , and i° are known. 

15. Assuming the functions of i° and 2 as known, compute the 
sines and cosines of 3 , 4 and 5 . 

(Suggestion. Apply the results of Problem 14.) 



ii3] FUNCTIONS OF TWO OR MORE ANGLES 223 

16. By the law of sines 

a _ sin A 

b sin B 
Taking this proportion by composition and division, we obtain 

a -f- b _ sin A -f- sin B 
a — b sin A — sin B 

Apply formulas (1), Art. 113, to this result and deduce the law 
of tangents. (This constitutes an independent proof of the law of 
tangents. This proof is the one generally given in elementary text- 
books on trigonometry.) 

17. From the law of sines we obtain readily 

c sinC c sinC 



a — b sin A — sin B a-\- b sin A + sin B 

Hence deduce the double formulas 

c = sin Hj+g) c _ = cos%(A + B) 

a- b sin \ (A — B) ' a + 6 cos J (4 — 5) 

18. In Fig. 83, let A' = angle PAC, B f = angle PBC, then by 
applying the law of sines to each of the triangles PAC and PBC, 

we find 

7 a sin .5' bsinA' , x 

6/3 = . = — » (1) 

sm p sm a 

from which 

sin^4 / __ a sin a 

sin -B' 6 sin (3 

By composition and division and the identity in Problem 6 

sin^' + sin 2?' __ a sin a -\- b sin g _ tan| {A' -j- i? r ) , >. 

sin ^4' — sin J3' a sin a — & sin j8 tan § (A f — B') 

Now ^4' + B' is known from the relation 

A' + B' + C + a + (3 = 36o°, 

and therefore (2) enables us to find A' — B' . Hence A' and B' 
may be found and with these known, the law of sines gives 

, & sin 04' + a) , a sin (B' + g) 

rfi = * > #2 = : — > 

sm a sm j8 

This constitutes another solution of the three-point problem. 



224 



PLANE TRIGONOMETRY 



[chap. XI 



19. Prove the formulas (1), Article 113, geometrically by means 
of Fig. 147. 

Suggestion. Take the radius of the arc equal to unity, that is, 
OE = OD= 1, then 

sin x + sin y = 2 • GF, 
sin x — sin y = 2 • LF, 
cos x + cos y = 2 • OG, 
cos x — cos y = — 2 • ZJ}. 

20. Show that 

tan (a -j- g) -f- tan g _ sin (g+2i) 

tan (a + #) — tan ic sin a 




CHAPTER XII 
TRIGONOMETRIC EQUATIONS 

114. Angles Corresponding to a Given Function. Every given 
angle has a single sine, cosine, tangent, etc. In Chapter X it was 
shown how these functions may be found from the tables by first 
expressing them in terms of an angle in the first quadrant. Suppose 
now that one of the functions of an angle is known and it is required 
to find the angle. If the angle is known to be less than R, it is at 
once found from the tables; if less than 2 R, there also is no uncer- 
tainty except in the case of the sine (and its reciprocal the cosecant), 
which may be either the angle given in the table or its supplement. 
But in case no restriction is imposed on the magnitude of the angle, 
the given function may belong to any one of an unlimited num- 
ber of angles. We shall show how all the angles corresponding 
to a given function can in each case be expressed by a single for- 
mula from which their separate values may be written down when 
required. 

115. Principal Value. Of all the angles which correspond to a 
given function, the one which has the least numerical value is called 
the principal value (see Article 81). If there are two least values 
with opposite signs, the positive angle is taken as the principal 
value. 

Thus, if sin 6 = J, = 30 or 150 or either of these values in- 
creased or diminished by any number of times 360 . Of all these 
angles, 30 being the least numerically, is considered the principal 
value of the angles whose sine is §. 

If sin 6 = — J \^3 ? 6 = — 6o°, — 120 or either of these values 
increased or diminished by any number of times 360 . Here — 6o°, 
having the least numerical value, is the principal value of the angles 
whose sine is — J_V 3. 

If cos 6 = J V2, 6 = 45 or — 45 or either of these increased or 
diminished by multiples of 360 . In this case 45 , not — 45 , is the 

principal value of 0. 

225 



2 26 PLANE TRIGONOMETRY [chap, xn 

116. Formula for Angles Having a Given Sine. 

Let it be required to determine 6 from the equation 

sin 6 = k. 
Let a = the principal value of 6. Then the other values of 6 are 



ir — a, 


— ir — a, 




2 7T + a, 


— 2 7T + a, 


/^~^\ /"""N 


ST — a, 


— 3 7T — a, 

— 4tt + a, 




47T+ «, 


v yv^ v 


ST — a, 


— Sir — a, 


^^^ v_^/ 


6ir -f- a 


— 6ir -\- a 


Fig. 148. 


etc. 


etc. 





It will be observed that the coefficients of ir are the integers 
1, 2, 3, 4, 5, 6, etc., - 1, - 2, - 3, -4,-5,-6, etc., 
while the sign of o; is positive or negative according as the coefficient 
of 7r is even or odd. Now (— i) n is always positive when n is an 
even integer, negative when n is an odd integer." Making use of 
this property, we can express the whole set of angles by the single 
formula 

6 = mr+ (- i) n a, 

where a is the principal value of the angles whose sine is k, and n 
any positive or negative integer. 

When n = o, we get the principal value of 6, for 

07r + (— i)°a — a. 

* 

117. Formula for Angles Having a Given Cosine. 

If cos d = k, and a the principal value of 6, the other values of 6 are 




Fig. 149. 



2 7r — a, 


— 2 7r — a, 


2tt -\- a, 


— 2 7r -f- a, 


4.7T — a, 


— 47r — a, 


4 7r + a, 


— 47T + a, 


6 ir — a, 


— 6x — a, 


6-7T + a, 


— 6 7T + Oi, 


etc. 


etc. 



120] TRIGONOMETRIC EQUATIONS 227 

All these values and no others are expressed by the single formula 

6 = 2 nir i a, 

where 2 n is necessarily an even number. 

118. Formula for Angles Having a Given Tangent. 

If tan 6 = k, and a the principal value of 0, the other values of 9 
are 

7T + Ci, — 7T + a, 

2 7r -f- a, — 2 x -f- a, 

3 tt + a, — 3 7T + a:, 

4 7r + a, — 4 7r + a, 
57T + <x, -57r + a, Fig. 150. 

etc. etc. 

All these values and no others are expressed by the single formula 

6 = nir -\- a, 
n being any positive or negative integer. 

119. Summary of Results. Summing up the results of the last 
three articles we have 




sin (nir + (— i) n a) = sin a, 
cos (2mr ± a) = cos a, 
tan (nir + a) = tan a, 



a) 



where a is the principal value of the angle, and n any positive or 
negative integer. 

These three formulas should be thoroughly understood and mem- 
orized by the student. 

120. Trigonometric Equations Involving a Single Angle. 

Equations involving one or more trigonometric functions of one or 
more unknown angles are called trigonometric equations. They may 
or may not involve other unknowns which are not angles. 

To solve trigonometric equations involving a single unknown 
angle, we express each of the functions which occur in the equation 
in terms of some one of them, and solve the resulting equation alge- 



228 PLANE TRIGONOMETRY [chap, xn 

braically, considering this function as the unknown. The cor- 
responding angles may then be easily found from the formulas of 
Article 119. 

Example i. Solve the equation 2 sin x + esc x = 3. 

Solution. Expressing the cosecant in terms of the sine, we have 

2 sinx + — — = 3. 
sinx 

Now consider sin % the unknown, call it s say, we then have 

, 1 

2 5+- = 3. 

Clearing of fractions and transposing, 

2^-35- 1 =0. 



01* 3 ± V2 2 — 4 • 2 x ± 1 i 

Solving 5 = ° ° — -=* — = ° =1 or f , 

4 4 

that is, sin x = 1 or \ 



11 



from which the principal values of x are x = - or - > 

2 6 

and by Article 119 the general values of x are 

x = nir + (— i) n -, or wx+(— i) n -- 
2 6 

The general values of x just written are equivalent to the two sets 
of values 



X 

x = -■> 

2 


5*~ 

7 
2 


QX 
2 


-3-^-, etc., 

2 


- 3 x 
> 
2 


— 7 X — 

7 
2 


IIX , 

, etc., 

2 


X 

X = -j 

6 


6 


I3X 
7 

6 


I7TT . 

— L - , etc., 
6 


— 7 X 

7 

6 


— IIX 

j 

6 


= I9T etc 

, CIV-., 





Check. If sin x — 1, esc x = 1, and 2 sin # + esc # = 2.1 + 1 =3. 
If sin x = J, esc # = 2, and 2 sin x + esc £=2. §+2 = 3. 

Example 2. To find in the equation tan + 3 cot = 4. 
Solution. Expressing the cotangent in terms of the tangent 
gives 



i2o] TRIGONOMETRIC EQUATIONS 229 

where t — tan 6. Solving for t, 
t 2 ~ 4 1 + 3 = o, 

; == 4±V 4 2 -4 l3 = 4^- 2 = 3o ri , 

2 2 

that is, tan0 = 3 or i, 

from which the principal values of 8 are 

= tan" 1 3 = 7 i 33 / 54" or tan" 1 1 = 45 . 
By Article 119 the general values of are 

6 = n '■ 180 + 7i°33 / 54" or » • 180 + 45 . 

Check. If tan = 3, cot = J, tan + 3 cot # = 3 + 3. i = 4. 
If tan 6 = 1, cot 6 = 1, tan + 3 cot 0= 1 + 3.1 = 4. 

Example 3. Find c/> in tan cf> + 2 v 3 cos <f> = o. 

• \* I — COS CD 

Solution. tan <f> = , 

cos <j) 



hence V 1 — cos 2 c/> + 2 V 3 cos 2 c/> = o. 

Transposing one term and squaring, 

I — COS 2 c/>= I2COS 4 c/>, 

1 2 cos 4 c/> + cos 2 c/> — 1 = o, 



1 ± Vj + 4 -12 ^_ — 1 ± 7 _ 1 
24 24 4 



cos 2 9 = — — = =— * = - or 7 



cos = I, -i, V-i -V-i 

The last two values of cos c/> are not admissible, since the cosine of 
a real angle cannot be imaginary, and the first two values each give 

c/> = 2 nir i - • 
3 

Check, cos (2 wx ± -]= J, tan ( 2 mr ± -)= ± V3, 

tan c/> + 2 V3 cos c/> = ±3 + 2 V3 • (J) = 6 for the lower sign 

but not for the upper. Hence the general solution of the equation 

tan 9 + 2 V 3 cos c/> = o is <j) =2nir — ~' 





230 PLANE TRIGONOMETRY [chap, xii 

This example clearly shows the necessity of verifying the results 
before accepting them as the solutions of a given equation. 

Example 4. Solve the equation sin x + cos x = V2. 
First solution. sin^+ Vi — sin 2 x = V 2, 

Vi — sin 2 x = V 2 — sin x. 

Squaring 1 — sin 2 # = 2 — 2 V 2 sin x + sin 2 x. 

Solving 2 sin 2 x — 2 v 2 sin x + 1 = o, 

2 V2 ± V8 - 4 • 2 V2 
sm x = — s — = > 

4 2 

* = Wr+ (— i) n -- 
4 

Check. sin(mr+(-i) n A = i V~ 2 , cos (nT+(-i) n -\ = ± i V2, 

according as n is even or odd. Therefore 

sin x + cos x = J V2 ± J V2 = V 2 or o 

according as n is even or odd. We see that the lower sign does not 
satisfy the original equation, that is n cannot be odd; hence the 
general solution of the equation 

sin x + cos x = V 2 

is x — 2 nw + J 7r. 

Second solution. We have given the solution which would most 
naturally suggest itself to the beginner. A more elegant solution of 
the foregoing equation is the following: 

Since cos x = sin (^ir — x), the equation to be solved may be 
written 

sin x + cos x = sin x + sin (J ir — x) = \/ 2. 

The middle member is the sum of two sines, which by Article 113 
may be transformed into a product, thus 

sin x + sin (J ir — x) = 2 sin J t cos (x — \ ir) = V 2, 
that is 2 • \ V2 • cos (x — J ir) = V 2, 

or cos (x — Jtt) = 1, 

from which x — \tt = 2 n ± o, or x =2 n-{- jir. 



i2o] TRIGONOMETRIC EQUATIONS 231 

Example 5. Solve the equation a sin (x + a) -f b cos (x + /3) = o, 
a, 6, a, /?, being known constants. 

Solution, sin (x + a) = sin x cos a + cos x sin a, 
cos (x-\- (3) = cos x cos /3 — sin x sin (3. 

Substituting these values in the original equation, 

a (sin x cos a + cos x sin a) + b (cos x cos /3 — sin x sin /3) = o. 

Collecting the coefficients of sin x and cos x separately, we have 
(a cos o: — b sin 0) sin x + (a sin a + b cos /3) cos x = o. 

Dividing by cos x, and solving for tan x, 

a sin a + 6 cos 3 

tan x = ; — - 1 

a cos a — b sin /3 

from which 

- _i /a sin a + & cos 3 

x = n-w — tan 1 ' - - 



f a sin a + & cos ff \ 
\ a cos a — Z> sin jS / 



Check. To check complicated results like the one just obtained 
it is best to assume arbitrary values for the constants. Thus if 

7 o o o ash\a-\- b cos 3 

a = 2, b = 1, a = 20 , p = 15 , ! r— ^ = 1.0181, 

a cos a: — £ sin /3 



x = W7r — tan x 1.0181 = W7r — 45 31 , 
x + a. — n-K — 25°3i / , x-\- ft = mr — 3o°3i / . 

Substituting these values in the original equation, we have 
2 (± 0.4308) + 1 (=F 0.8615) = o. 

Exercise 52 

In the following find both the principal and the general value of 
the angle: 

1. tan0 = 2 sin 6. 

Ans. 6 = mr or 2 rnr ± -; principal values o and -• 

3 3 

2. 3 sin 2 x = cos 2 x. 

Ans. x = mr ± - ; principal value - • 
6 6 

3. 3 tan 2 x — 4 sin 2 x = 1. 

Ans. x = mr ± '-; principal value -• 



232 PLANE TRIGONOMETRY [chap, xn 

4. 2 sin 2 cf> = 3 cos (f>. 

Ans. 6 — 2 mr ± -; principal value -• 
3 3 

5. tan 3/ + cot 3/ = 2. ^4ws. y = Tfi + iir; principal value -• 

4 

In the following problems find those values of the variable angle 
which are less than 360 : 

6. tan 2 x+ csc 2 x = 3. ^4ws. x=-> — > . a2 — > ■*— • 

4 4 4 4 
_ ~ 1 



7. sin (9+ cos0 = 1. Ans. 6 = o, Jtt. 

8. esc 



A /- , 7T IIJ 



7T 5X 

9. sin 2 t — 2 cos t + i = o. i4»s. / = ~'~' 

10. 3 sec 4 — 10 sec 2 + 8 = o. 

A a IT IT 37T 57T 77T 5 7T 7^ 1 1 7T 

^4/w. 6 = -> - } iL - ? ^— » *— ? - — > — j — - — 
6446644 6 

11. tans: + sec 2 x = 7. 

4»s. x = 63 26' 04", io8°26 r 06", 243 26' 04", 288 26' 06". 

12. 6 cos 2 x + 5 sin x = 7. 

i4*w. *= i9°28 / i6 // ,3o°, 150 , i6o°3i , 44 / ;. 

13. sinx+ csca: = f. ^4ws. # = 30 , 150 . 

14. sin x — cos x = f . ^4ws. No solution. 



T , - , , ,, , . ac ±b\^a 2 + b 2 — c 2 
i<. Ii asmx-\- a cos x = c, show that sin x = — — -• 

3 ' a 2 + & 2 



16. If a tan # + b cot # = c, show that tan # = 



c ± v c 2 — 4 afr 
2 a 



17. If sin (a + x) = m sin #, show that sin x = =., 

\/m 2 — 2 m cos a + 1 

t m — cos a 

or more simply cot x = : 

sin a 

18. If tan (a-\- x) = m tan x, show that 

. _ (m — i)± V (m — i) 2 — 4 w ta n 2 a? 
tan x — — • 

2 m tan a 

19. If tan (a + x) tan # = m, show that 

, _ — (1 + m) tan « ±^( 1 -1- w) 2 ta n 2 a j- 4 m 

tan x — • 

2 



121] TRIGONOMETRIC EQUATIONS 233 

By expressing each product as a sum or difference of functions 
show: 

20. If sin (a ± x) sin x = m, then cos (a ± 2 x) = cos a^2w. 

21. If cos (a ± x) cos x = ra, then cos (a ± 2 x) = 2m — cos a. 

22. If sin (a ± x) cos x = ra, then sin (a ± 2 x) = 2m — sin a. 

23. If cos (a ± x) sinx = m, then sin (a ± 2 x) = sin a ± 2 m. 

121. Trigonometric Equations Involving Multiple Angles. 

When an equation involves multiple angles, as the equation 

cos 3 + sin 2 = a, or tan 2 x = cot 5 x, 
it can frequently be solved by two or more different methods, and the 
answer will appear in various forms according as one or the other of 
these methods has been employed in the solution. Generally the dif- 
ferent forms of the answers can easily be identified, but in some cases 
considerable ingenuity is required to show that the different forms are 
really the same. Thus it is easy to see that 2 mr ± § ir and mr + J 7r 
(n being any integer, but not the same integer in both forms) ex- 
press the same general value, but it is not so easy to see that 
d = n-K-\-\-K or W7r+ ( — i) n sin _1 | (— i±V / 5) 

and 9 = 2 mr — %7r or 1 

5 10 
are equivalent results. 

Example i. Solve the equation sin 2 $. = cos 9. 

First solution. Substituting for sin 2 9 its value in terms of the 
single angle 9 (Article in, (1)), we have 

2 sin 9 cos 9 = cos 9. 

Transposing and factoring, 

cos# (2 sin0 — 1) = o, 

from which 

cos = o, or sin = \. 

Therefore 

= 2 tit ± \ t = n-K + \ 7T*, or mr + (— i) n - • (1) 

6 

* 9 = 2 n-w + \ ir and 2 mr — \tt = (2 n — 1) w -j- \-k. Now 2 n represents 
every even and 2 n — 1 every odd number, hence we may write 6 = mr + \ir, 
where n is any integer, even or odd. 



234 PLANE TRIGONOMETRY [chap, xii 

Check. 

sin 2 = sin (2 mr + x) = o, or sin (2 mr +(— i) w -)= ± J V3, 
according as n is even or odd. 

cos0 = cos (W + Jx) = o, or coslnw +■(— i) n -J = ± \ V3, 

according as « is even or odd. 

Substituting these values in the original equation, we have 

o = o, or ± \ V3 = ± J V'3. 
Second solution. 

Since cos = sin (J 7r — 0), we have 

sin 2 = sin (Jx — 0), 
from which 2 — nir + (— i) n (J 7r — 0), 

or = - — L -^ ' z • (2) 

2+(-l) n 

TTwrrf solution. Transposing the second member of the equa- 
tion, we have 

sin 2 — cos = sin 2 — sin (J 7r — 0) = o. 

By Article 113, the difference of two sines may be tranf-ormed 
into a product of a sine and cosine, thus 

sin 2 — sin (J x — 0) = 2 cos (§ + J x) sin (f — J x) = o, 

from which 

cos (J + i x) = o, or sin (f — J x) = o, 

and | + | X = 2»T±§X = »X+^X, Orf0— Jx = WX, 

that is, = 2 mr + J x, or § mr + J x. (3) 

Identification of results. 

If in (2) » is odd, say 2 m -\- 1, we have 

0= (2 ra + i)x — Jx= 2 mx + |x ? 
and if w is even, say 2 m, we have 

z, 2 mx + |x 2 11 

3 
This -shows that the results (2) and (3) are the same. 



i2i] TRIGONOMETRIC EQUATIONS 235 

Next consider the second value in (3). Every integer n is either 
some multiple of 3, say 3 m, or some multiple of 3 increased by unity, 
say 3 m + 1, or some multiple of 3 increased by 2, say 3 m + 2. 

In the first case, n = 3 m, we have from the second value of (3) 

2 flTT 1 7T 2.X MIT 1 7T , 7T 

1 — = — ^ 1 — — 2 mir -\ 

3636 6 

In the second case, n = 3 m + 1, 



2 w 



7T|7r 2 (l W + i) 7T , 7T / , n 7T 

— h - = — ^ ! — r - = (2 w + 1) 7r — - • 



36 3 6 6 

In the third case, n = 3 w + 2, 



2 W7T 



,7T 2 (^ W + 2W , 7T / , \ ,7T 



3636 2 

This last value, combined with the first value of in (3), gives 
0=2 nir+^ir, or (2 w+i)7r+j7r, that is, 6 = mr-\-^ir in even or odd), 

and 

6=2 mir +|f, or (2 m + 1) tt — J 7r,. may be written 

0=W7T+(- l) n ~- 

v J 6 
This shows that the results (3) and (1) are the same. 

Exercise 53 

Find the general value of the angle in each of the following equa- 
tions : 

i; cos 2 x + cos x + 1 = o. Ans. x — nw — J x, or 2 mr ± f x. 

, -1 2 WX , X 

2. cos 5 # = sin 4 x. ^Itw. x = 2 raj — f x, or h — 

9 18 

, 2 WX 

3. cos 5 x = cos 4 x. ^Lws. x = 2 wx, or 

4. sin 4 # = sin 5 x. ^4ws. x=2nir, or (2^+1)-- 

5. tan 50= cot 2 0. ^4ws. = 1 

7 *4 

6. sin 3 x + sin 2 x + sin x = o. 

27T 
.4 WS. X = 2 WX, Or 2 W7T ± I 7T, Or 2 W7T rh 



236 PLANE TRIGONOMETRY [chap, xn 

7. cos x — cos 3 x = sin 2 x. 

Ans. x = mc, or wx — |x, or mr -\- (— i) n ~- 



8. cos 5 3; — cos 3 y + sin y = o. 

^4ws. y = nir, or J wx + (— i) n — . 

24 

9. cos (6o° — x) + cos (6o° + #) = -J. 

^4ws. a; = 2 mr ± cos -1 \ = 2 n i8o°± 70 32'. 

10. cos mx = sin kx, m and k being known. 

A in-w + \ x) 

ym ± &) 

11. cot^> = tan&0, & being known. 

& + I 2 (& + i) 

12. tan 2 x tan x — 1. ^4/w. # = nx ± -• 

6 

Solve each of the following problems by each of three methods, 
and identify the results. 

13. cos 26 = sin 6. Ans. 6 = 2 nw — Jx, orf nir -\ 

6 



14. cos 3 x = sin 2 x. Ans. x = 2 wx — J x, or f mr -\ 

10 

Suggestion. \ (- 1 + V5) = sin — , i (- 1 - V5) = sinf- 2Z\ 

10 \ 10/ 

15. sin 3 x = cos 2 #. ^4w5. # = 2 mr + J x, or § mr H 

10 

122. Trigonometric Equations Involving Two or More Vari- 
ables. When there are given two or more trigonometric equa- 
tions, involving two or more variables, the solution, if it is possible 
at all, generally depends upon more or less ingenious combinations 
of the equations, for which no definite rules can be given. We shall 
illustrate the various methods and devices commonly employed by a 
few examples chosen from among those which most frequently occur 
in applied trigonometry. 



122] TRIGONOMETRIC EQUATIONS 237 

Example i. To find r and 6 from the equations 

r sin 6 = a, (1) 

r cos 6 = b, (2) 

a and b being known constants. 

First solution. Dividing the first equation by the second, we have 

tan 6 = - 
b 

from which 6 may be found, r is then found from either (1) or (2), 

a b 

sin 6 cos 

Since the angles 6 as determined from the tangent differ by multiples 
of it, sin 6 and cos 6 will have two values each which are numerically 
equal but opposite in sign. Therefore r will have two values which 
are equal and opposite in sign. If from the outset it is known that r 
can be positive only, which is the case frequently, then sin 6 must have 
the same sign as a and cos 6 the same sign as b. 6 must then be 
limited to the quadrants determined by these signs. 

Second solution. Squaring each equation and adding the results 

gives 

r 2 = a 2 + b 2 , 

from which r is obtained. Then 6 is found from either of the equa- 
tions 

. Q a a b 

sin 6 = -> cos = - 
r r 

Of the two methods the first is better adapted to the use of loga- 
rithms than the second. 

Example 2. Find r, 6 and from the equations 

r cos d cos cf> = a (1) 

r cos 6 sin <fi = b (2) 

r sin 6 = c. (3) 

First solution. Dividing the second equation by the first we 

obtain 7 

tan <p = - 
a 



238 PLANE TRIGONOMETRY [chap, xn 

from which <f> is obtained. <j> being known, r cos is obtained from 
either equation (1) or (2) 

rcosd= -2— = — — • (4) 

cos 9 sin 9 

From (3) and (4) r and may be found as in Example 1. 
Second solution. Squaring each of the equations (1), (2) and (3) 
and adding the results gives 

r 2 = a 2 + b 2 + c 2 

from which r may be found. 

r being known, is found from the equation (3), and, r and being 
known, <^> is found from either equation (1) or (2). 

The first solution is preferable if logarithms are to be used through- 
out. 

Example 3. Solve the equations 

r sin (a + 6) — a 
r sin (0 + 6) ) = b 
for r and 0. 

Solution. Applying the addition theorem for the sine, we have 

r sin a cos + r cos a sin = a 
r sin /3 cos -\- r cos /3 sin = 5. 

Put r cos = x, r sin = y, the equations then become 

x sin a + v cos a = a 
x sin |8 + y cos (3 = b. 

Solving these equations for x and y, we obtain 

„ a cos B — b cos a a cos 8 — b cos a 

# = 7- COS = ; ■ = 7^—, : • 

sin a cos (3 — sin a sm /3 sin {a — p) 

. n a sin /3 — b sin a & sin a — a sin fl 

y = rsmv = : — - : = ; — — -■ 

cos a sm /3 — sin a cos /3 sm {a — p) 

x and y, that is, r cos and r sin being known, r and are found as 
in Example 1. 

Exercise 54 

In the following equations, consider r positive. 

1. Given r sin = 8.219, r cos = 12.88, find r and 0. 

Ans. r = 15.28,0 = 32°33 / - 

2. Given r sin = 3, r cos = 4, find r and 0. 



i2 2] TRIGONOMETRIC EQUATIONS 239 



3. Given r sin 6 = 27.138, r cos 6 = — 92.692, find r and 6. 

Ans. r = 96.583, 6 = 163 40' 52". 

4. Given r cos cos <£ = 59.953, 

r cos# sin <f> = 197.207 V Find r, 6 and <^. 
r sin<f> = 39.062. 
^4ws. r = 208.16, = io° 49' 00", <f> = 74 42' 00". 

5. Given r sin 6 cos = 5,! 

r cos sin (f> = 12, !■ Find r, 6 and <£. 
r sin 6 = 84. j 
Ans. By natural functions, r = 85, 6 = 8i° 12', <^> = 86° 36'. 

6. Find r, X and ju from the equations 

r cos X cos fi = 4, r cos X sin p = 5, r sin X = v 59. 

7. r sin [-+«)= V 3, r sin (- + «)= 1. Find r and x. 

^4ws. r = 2, * = o°. 

8. Show how to solve the equations 

r cos (x — a) = a, r sin (x + j8) = 6, for r and x. 

9. Solve the equations 

cos {x — y) = J V2, sin (x + y) = | V3, 

giving both the principal and general values. 

^4^?. Principal values, x = * — or — > y = — or - — 

24 24 24 24 

General values, x = (m + 2 »)- +(— i) OT - ± -> 

2 6 8 

,='(.- a «)*+(-i)-lT|, 

2 6 8 

where w and n are any two integers. 

10. Solve the equations 

cos 2 x — cos 2 y = a, cos x — cos y = b. 

^.WS. X = 2 «7T ± COS -1 I — ! , 7 = 2 W7T ± COS X — 

\ 4b 2 / V 40 

11. Show how to solve the equations 

tan (x + y) = a, tan x • tan y = b. 



240 PLANE TRIGONOMETRY [chap, xii 

123. Solutions Adapted to Logarithmic Computation. The 

solution of a problem in trigonometry is not considered completed 
until it can be effected by the use of logarithms, in fact the adapta- 
tion of formulas to the use of logarithms forms an important part in 
trigonometric investigations. Many trigonometric equations whose 
algebraic solution is exceedingly simple require further treatment 
from the trigonometric point of view. Equations 15 to 19, Exer- 
cise 52, are typical equations of this kind. We will show now how 
each of these equations may be solved by logarithms. 

Example i. To solve the equation 

a sina?+ b cos a? = c. (1) 



Solution. Divide each term of the equation by v a 2 + b 2 , then 

the coefficients — =■ and ■ — =- are fractions the sum of whose 

V a 2 + b 2 vV + b 2 

squares equals 1, we may therefore put 

: = cos <f>, — _ = sin <b, (2) 

vV + b 2 Va 2 + b 2 

and the given equation becomes 

sin x cos (f) + cos x sin <f> — 



Va 2 +b 2 



or 



sin (x + <t>) = . = - cos (f>= - sin ^>, (3) 

V a 2 + b 2 a 



and from (2) 



tan $ = -> (4) 

a 



Having found <j> from (4), x is found from (3). 
An angle like the angle </>, introduced to facilitate the solution of 
a problem, is called an auxiliary angle. 

Numerical Illustration 
Suppose the given equation is 

3.4537 sin x - 0.9328 cos x = - 1.3794, 
then a = 34537, b= - 0.9328, c = - 1.3794. 



123] TRIGONOMETRIC EQUATIONS 24 1 

Solution. 

By (4) By (3) 

log b — 9.96979 n * log c = 0.13969 n 

log a = 0.53828 cologa = 9.46172 

log tan (f> = 9.43 1 51 n log cos </> = 9.98471 



cj>\ = — 15° 06' 52", log sin (x+ <f>)= 9.58612 n 

X + (f> = nir + (- j)n ( 22 ° 40 ' 4 g//) 

^ = wx+i5°o6 , 52 ,, + (- i)»+i (22°4o , 4 8 ,/ ) 

The two smallest positive values, {n = 1, 2), are 

x = 217 47' 40", 352 26' 04". 

Check. By (3) logc = 0.13969 n 

colog b = 0.03021 n 

log sin <f) = 9.4162 1 n 

log sin (# + <j>) = 9.5861 1 n 

Example 2. To solve the equation 

a tan oc + b cot 05 = c. 

Solution. Expressing the tangent and cotangent each in terms 
of the sine and cosine, the equation reduces to (Problem 19, Exer- 
cise 50) 

c sin 2 x + (a — b) cos 2 x = a + b» 

This equation is of the form 

a sin x -\- b cos # = c, 

which has been solved in Example 1. 

Example 3. Solve the equation 

sin (a + a?) = m sin x, (1) 

Solution. By composition and division the proportion 

sin (a -j- x) _ m 
sinx 1 

gives rise to 

sin (a -f- x) -j- sin x _ 2 sin (a-\- %x) cos | o: 
sin (a + #) — sin x 2 cos (a + \ x) sin J a 

4- / 1 1 \ + 1 w+ 1 
= tan (a +f #) cot f a = > 

w — 1 

* This n indicates that the number to which the logarithm belongs is negative. 
f Since 4> is an auxiliary angle which is not retained in the end, any one of its 
values may be used. 



242 PLANE TRIGONOMETRY [chap, xii 

or tan (a + J x) = — ^— tan J a • (2) 

m — 1 

From (2) a + \ x and hence x can be found. 
If m = tan <£ 

ra+i tan^) + tan|7r , / j ■ 1 \ , / ± 1 x 

= : \ I , = - tan (0 + f tt) = cot (<j> - Jir), 

m — 1 1 — tan \ ir tan <fi 

hence 

tan (a + | #) = cot (</> — J 7r) tan § a, where tan <j> = m. (3) 

Many computers prefer (3) to (2) in solving equation (1). 

Example 4. Solve the equation 

tan (a-{- x) = m tan x. 

Solution. Taking the proportion 

tan (a -j- x) __ w 
tanx 1 

by composition and division, we obtain 

tan (a -\- x) -\- tan x _ m -f- 1 
tan (a + x) — tan x w — 1 

But by Problem 20, Exercise 51, 

tan (a + x) + tan # _ sin (o; + 2 #) 

■ ; — ; — : — : > 

tan {a + x) — tan x sin a 

hence 

sin (a+2i)= — ! — sm a, (1) 

m — 1 

or we may write the result in the form 

sin (a-\- 2 x) = cot ((f) — \ t) sin a, where tan cf> = m. (2) 

Either (1) or (2) may be used to find a -\- 2 x and hence x. 

Example 5. Solve the equation 

tan (a + x) tan x = m. 

Solution. Expressing the tangents in terms of sines and cosines, 

we find 

sin (a + x) sin x = m cos (a + x) cos x, 

which by Problem 10, Exercise 51, may be written 

cos a — cos (a + 2 x) = m [cos (a + 2 x) + cos a], 



123] TRIGONOMETRIC EQUATIONS 243 

from which 

/ ' , n 1 — m , n 

cos (a + 2 x) = cos a. (i) 

1 + m 

By Example 3, — ^-— = cot ((f)— \ic), where tan <f> = m, 

m — 1 

hence = tan ((f) — Jx), 

w + 1 

and = — tan (<f> — \ if) = tan (\ ir — <j>), 

1 + ni 

so that equation (1) may be written 

cos (a + 2 #) = tan (J 7r — 0) cos a, where tan (f> = m. (2} 

From (1) or from (2) a + 2 x may be found and hence x. 

Exercise 55 

1. If cos (x + d) = m cos x, show that x is given by the formula 

tan (a + \ x) = tan (J w —(f)) cot J a, where tan (f) = m. 

2. If sin (x -$- a) — m cos #, show that x is given by the formula 
tan (\a-\- \ir -\- x) = cot (J ir — <£) tan (J 7r — J a), where tan (f>= m. 

3. If cos (x -\- a) = m sin #, show that x is given by the formula 
tan (\ a — \ir -\- x) = tan (\ w — (f)) cot (| ir + § a), where tan </> = m. 

4. If tan (# + a) = w cot x, show that 

cos (a + 2 #) = tan (\ir — (f)) cos a, where tan (f) = m. 

5. If cot (x + a) = m cot (# — a), then 

sin 2 # = cot (<f> — \ ir) sin 2 a, where tan (f) = m. 

6. If tan (a + #) cot x = m, show that 

sin (a + 2 #) = cot (</> — i tt) sin a, where tan (f) = m. 

7. If tan (a + #) tan (a — x) = m, show that 

COS 2 X = COt (| 7T — </>) cos 2 a. 

8. Find the angles between o° and 360 which satisfy the equation 

4 sin x + 3 cos # = 5. 

Ans. x = 53 07' 45". 

9. Find from the equation 

2.76 cos — 2.32 sin 6 = 1.91. 

Ans. d = if 59.6', 261 55'. 



244 PLANE TRIGONOMETRY [chap, xn 

10. Find the general solution of the equation 

V 3 sin x — cos x = V 2. 



7T 



Ans. x = 2 nir -\- I 5 ? ir, or (2 n -\- 1) T — 

12 

11. Find the general solution of the equation 

(1 + V 3) tan # + (1 — V3) cot a; = 2. 
1 1 t 

X = WK -+- f 7T, W7T 

12 

12. Find r and # from the equations 

r sin (a + x) = m, 
r sin (fi -\- x) = w. 

Suggestion. Form the sum and difference of these equations, and 
change each into a product by the formulas in Article 113. Divid- 
ing the first result by the second gives 

ta „ Ai+J + x )= VL±JL tan^Lni 
\ 2 ) m—n 2 

tan [ ■ + x ) = tan (| x + <£) tan - — -> where tan <f> = —. 

\ 2 } 2 m 

13. If r cos {a -f- x) = m, r cos (ft + x) = n, then # is given by 

cot ( ^ + x ) = tan (i 7r + 0) tan — > where tan = — . 

\ 2 / 2 w 

14. Adapt the formula 



or 



w sm o; 
tan x = 



1 + m cos a 
to computation by logarithms. 

Suggestion. Express the tangent in terms of the sine and cosine 
and clear fractions, the result may be written 

sin x = m sin (a — x), 
which by Example 3 may be transformed into 

tan (x — I a) = tan \ a = tan (<j>— Jt) tan J a, where tan <j> = m. 

m + 1 

15. Show that 

a + b = j- where tan 2 = - > 

cos 2 9 a 

a — b = a • cos 2 </>, where sin 2 cj> — — 

a 



124] 



TRIGONOMETRIC EQUATIONS 



245 



These formulas enable us to find the logarithm of any sum or 
difference. Thus, 

log (a + b) = log a + 2 colog cos cp, log tan <f> = \ (log b — log a). 
log (a — b) = log a + 2 log cos <£, log sin <£ = \ (log & — log a), 

16. Establish the following transformations: 

■ 7 • a cos (a — 0) 
a sin a + 6 sin a = 

COS0 



a cos 



7 • a cos (a + 6) 
a — sin a = — ! — - 



COS0 



where tan 6 = — 
a 



124. Inverse Functions. The two expressions 

y= sin oc (1) 




a? = sin -1 ?/ 



(2) 



represent different views of the same relation. 
Fig- I S I - Xhe first one states that y is the sine of x, the 

second that x is the angle or arc (measuring the angle) whose sine 
is y. (1) expresses y in terms of x, (2) expresses x in terms of y. 
Taking the sine of each member of (2) gives us 

sin x = sin (sin -1 ^) = y, by (1), (3) 

and taking the inverse sine of each member of (1) gives us 

sin -1 y — sin -1 (sin x) = x, by (2). (4) 

In precisely the same way it may be shown that 

cos (cos~ 1 x)= x, cos -1 (cos#) = X, 
tan (tan -1 x) = x, tan -1 (tan x) = x, 
etc., etc. 

From these relations it appears that of each pair of operations, 
say for example that of taking the sine and that of taking the arc- 
sine, either undoes the other, that is, if the two operations are per- 
formed in succession, the result is that the quantity operated on is 
left unchanged. This explains why sin -1 # is called the inverse sine 
of_ x, tan -1 x the inverse tangent of x, etc., in fact in any pair of 
such functions each is the inverse of the other. Viewed as opera- 
tions, the relation of the members of each pair is like that of 
addition to subtraction, of multiplication to division, of involution to 
evolution. 



246 PLANE TRIGONOMETRY [chap, xn 

In general, if y = / (x) represents any function of x, the inverse 
function is represented by/ -1 (x), and the relation between the two 
is always such that, considered as operations on x, each undoes the 
other, that is, 

/l/" 1 (»)] = «, and /- 1 [/W] = «. (5) 

If y = / (x) is known, /~ l (x) is found by solving (provided we can 
solve) y = f(x) for x, and by substituting in the result x for y. Thus, 
if 

y = J^L±=f( x ) 9 ( 6 ) 



we find x = V3 3; + 4 = / _1 (3;), 

and V 3 x + 4 = f~ l (x). (7) 

To verify the relations (5) we substitute for x in (6) the expression 
(7), and for x in (7) the expression (6); thus, 

3 

There is this important difference between the trigonometric func- 
tions and their inverses, — while each of the former has a single 
value, each of the latter has an indefinite number of values. Thus, 

if x = -, sin x has a single value, namely J, but if x = J, sin -1 x 
6 

may have any of the values -, — , -2 — , -Z_ } etc. In general, 

6 6 6 6 

sin -1 x = W7r + (— i) w a, 
cos -1 :*; = 2 nir ± a, 
tan -1 x = W7r + a, 

where a is the principal value of the angle. 

Any relation between trigonometric functions may be expressed 
by means of inverse functions. We shall illustrate the method by 
some examples. 

Example i. Express the relation 

cos 2 A = 1 — 2 sin 2 A 
in terms of inverse functions. 



i2 4 ] TRIGONOMETRIC EQUATIONS 247 

Solution. Let sin A = m, then A = sin -1 m } and the given equa- 
tion becomes 

cos (2 sin -1 m) = 1 — 2 m 2 , 

or 2 sin -1 m = cos -1 (1 — 2 m 2 ). 

Example 2. Express the formula 

sin (.4 + B) = sin ^4 cos 5 + cos A sin 5 
in terms of inverse functions. 

Solution. Let sin A = m, sin B = ft, 



then cos A = V 1 — ra 2 , cosi? = v 1 — ft 2 , 

and the given formula becomes 

sin (sin -1 m + sin -1 ft) = m V 1 — ft 2 + n Vi — w 2 , 
or sin -1 w + sin -1 ft = sin -1 (m V 1 — n 2 + ft V 1 — m 2 ). 

Example 3. Express the formula 

r A 1 v \ tan A + tan B 

tan (/I + B) = 



1 — tan A tan J3 
in terms of inverse functions. 

Solution. Let tan A = m, tan B = n, 

then A = tan -1 m, B = tan -1 ft. 

Substituting 

tan (tan x m + tan x 7?) = ! , 

1 — mn 

+ -1 I 4- -1 4- -1 ^ + n 

or tan 1 m-t-tan i ft=tan x ! 

1 — mn 

Formulas involving inverse functions may be verified by reversing 
the process illustrated in the above examples. 

Example 4. Show that 

A- -1 I 4- -1 A- -1 m + n 

tan l m -\- tan l n = tan l 

1 — mn 

Solution. Put tan _1 w = A, tan -1 ft = B, 

then m = tan A , n = tan B. 

Substituting, we find 

A ■ „ _, tan ^4 + tan 2? 

^4 + B = tan * L - > 

1 — tan A tan i> 



248 PLANE TRIGONOMETRY [chap, xn 

or 

4. ( a 1 t>\ tan A + tan B 
tan (^4 + B) = • — -• 

1 — tan A tan B 

This latter expression we know is true, hence the original expression 
is also true. 

Example 5. Find the value of 

tan -1 J + tan -1 J. 

Solution. Put tan -1 ! = ^4, tan -1 ! = B, 

then tan A = !, tan B = |, 

and the given expression becomes A + B. 

at 4. / a 1 r>\ tan ^4 + tan J5 i + 3 

Now tan (A + B) = ! = — \ , — 1, 

1 — tan A tan B 1 — 2X3 

therefore A + B = tan -1 ! + tan -1 1 = mr H 

4 

Example 6. Solve the equation 

sin -1 2 x-\- sin -1 3 # = cos -1 (— |). 
Solution. Put sin -1 2 # = A, sin -1 3 # = B, 

then sin ^4 = 2 £, sin B = 3 #, 



cos ^4 = Vi — 4 # 2 , cos 5 = vi — 9 # 2 , 
and the given expression becomes 

A + B = cos -1 (- f). 
Take the cosine of both sides, 

cos (A + B) = cos A cos B — sin ^4 sin B = — § , 
and expressing this in terms of x, 

Vi — 4X 2 ' VI — 9 £ 2 — 2 X • 3 £ = — f. 

Solving for #, 



x — I3. 

Exercise 56 
1. Find the general value of each of the following angles: 

/ v 2 / — 

sin -1 J, cos -1 , tan -1 V 3, cos -1 o, sec -1 1, tan -1 00 . 



2 



^4ttS. W7T + (— i)"-, 2W7T±-, «7T-f--,etC. 
6 4 3 



124] TRIGONOMETRIC EQUATIONS 249 

Considering principal values only, verify the following : 

nr 



2. tan x § — tan 1 }=- 

4 



3. tan -1 m-\- tan -1 — = -• 

m 2 

4. COS _1 r 8 7 + cos -1 if = ~ * 

2 

5. tan-4 + tan- 1 i + tan" 1 T V=-- 

4 

Express the following formulas in terms of inverse functions : 

*> . a 2 tan 6 

6. tan 26= 

1 - tan 2 

Ans. 2 tan _1 ra = tan -1 ( ), where m = tan0. 

\ 1 — m 2 ) 

7. cos (A — B) = cos A cos B + sin A sin B. 

Ans. sin -1 m — sin -1 n = cos -1 (w« + v (1 — w 2 ) (1 — w 2 ), 

where w = sin A, n = sin £. 

8. sin 2 x = 2 sin x cos x. 

Ans. 2 sin -1 m = sin -1 (2 w v 1 — w 2 ), where w = sin #. 

9. sin 3 # = 3 sin x — 4 sin 3 x. 

Ans. 3 sin -1 w = sin -1 (3 m — 4 ra 3 ), where m = sin '#. 

10. Show that 

sin -1 w = cos -1 V 1 — m 2 = tan * 1 = — sec 1 

Vi-r V 1 — m 2 

-1 1 rf-i^ 1 
= CSC * — = cot l 



tnr 



m m 

11. Show that 

sin- 1 f + cos- 1 H = tan- 1 |f. 

12. Show that 

2 tan -1 \ + 3 tan_1 3 = tan -1 (— 3). 

Find x in the following equations : 

1 . • 1 7T i , v 21 

13. sin -1 2 #+ sin 1 # = -- Ans. x=± 

3 14 

14. tan -1 (1 + x) +tan _1 (1 — x) = tan -1 — • Ans. x = ± 5. 

2 5 

/- V^ 

15. sin -1 * + 2 cos -1 * = tan -1 V3. Ans. x = ± — *. 



250 PLANE TRIGONOMETRY [chap, xn 

16. Show that if / (x) = °^^ , then f' 1 (x) = — ^— . 



x 



17. Prove that the inverse of x 2 + 4 is V / x — 4. 

18. What is the inverse of logi £ ? Of 1 — x ? 

Ans. 10*, 1 — x. 

X ~~\~ T 

19. Prove that is its own inverse. 

x — 1 

Find the inverse functions of each of the following, and verify the 
results : 

20. f(x) = 

3 

2i./<y)-£±i*. 



22. / (0) = ^±_*. 4»5. /- 1 (0) = 2 \/£±-£. 

125. Review. 1. (a) Define the sine, cosine and tangent of any 
angle, (b) Give the signs of the principal functions in each of the 
four quadrants, (c) Give the formula which expresses the periodicity 
of the sine, (d) Prove that tan (d + nir) = tan 6. (e) Which other 
function has the same period as the tangent ? 

2. (a) Follow the changes in the sine of an angle as the. angle 
increases from o to 2 r. (b) Do the same for the cosine, (c) Do the 
same for the tangent, (d) Follow the changes in the reciprocal of 
the tangent. 

3. (a) Draw the lines which represent the various functions of an 
arc of a circle (or of its angle at the center of the circle) when the 
radius of the circle is taken as the unit of measure, (b) Draw these 
lines for an angle in the third quadrant, (c) Explain the derivation 
of the words secant, tangent, sine and cosine. 

4. (a) Prove geometrically that sin (R + 6) = cos 6, tan (3 R — 6) 
= cot 6, 6 being an angle in the first quadrant, (b) Prove that 
sin ( — 6) = — sin 6, cos (2 R + 6) = — cos for every value of 6. 
(c) Give from memory the principal functions of 30 , 150 , 210 , 
330 . (d) Give from memory the principal functions of 45 , 13 5 , 



o o 

225 ,315 ■ 



5. (a) Show that 
sin (— A) + cos (— A) _ sin (90 + A) + cos (270 — A) 
tan (-,4)- cot (- A) " cot (180 + A) + tan (360 - A) 



125] TRIGONOMETRIC EQUATIONS 251 

/n c . ,., sin (R + x) cos (R — x) , sin (R — x) cos (R + x) 

(b) Simplify * v H . ,'. —. -• 

cos (2 it + x) sm (2 it + x) 

(c) Find from the tables sin 234 , cos 342 , tan 134 54', sin 967 45', 
tan tnir -\- — 

6. (a) Prove the addition theorem for the sine, for the cosine and 
for the tangent, (b) Give the formulas for sin 2 6, cos 2 6 and tan 2 0. 

(c) Show that sin (x + 3/) sin (x — y) '== (sin x + sin y) (sin x —sin y). 

(d) Given cos x = f, find sin -, cos - and tan - without the use of 

22 2 

tables. 

7. Given the law of sines, prove the law of tangents. 

8. Express each of the following as a product: 



sin 7 + sin 15 , cos cos *—, sin A + sin 3 A + sin 4 A, 

2 2 



X 7. X 
COS Q — 

2 2 

9. By multiplying each side of the expression 



5=1 + cos x + cos 2 x + cos 3 x + . . . + cos fix, by 2 sin - 

2 

and expressing each product on the right as a difference of two sines 

show that 

. (2 n + 1) x , .x . (w + 1) x (wx) 

sm ^ ! — '- h sm - sin - — ! — — cos - — L 

~ _ 2 2 _ 2 

. /x\ • (x 

2sm U sin u 

10. (a) What is meant by the principal value of an angle ? 

(b) Give the principal values of sin -1 §, sin -1 — J, cos -1 J, tan -1 — 1. 

(c) Give the general values of the angles under (b). 

n. (a) What is meant by a trigonometric equation ? (b) Solve 
the following equations, sin x = — cos x, 2 sin 2 x + 3 cos 2 x = 5, 
a sin x = b tan x. 

12. Solve the equations: 

(a) sin (x + c) — cos x sin c = cos c. 
(5) sin (a — x) = cos (a + x). 

V2 
(c) sin (x + y) = cos (x — y) = 



252 PLANE TRIGONOMETRY [chap, xn 

13. Solve the equations: 

(a) sin 2 y -\- sin 3 y = 3 sin y. (b) 1 + cos x = cos - • 

2 

(c) sin x cos x = J. (J) sin # + cos # = V 2. 

14. (a) Show how to adapt the equation a sin x + 5 cos # = c 
to solution by means of logarithms, (b) Solve the simultaneous 
equations- : x cos a -{- y sin a = a, x sin a + v cos a = 6. 

15. (a) Define the inverse trigonometric functions. 

(b) Complete the following equalities, sin" 1 x = cos -1 ( ) 

= tan" 1 ( ) = sec -1 ( ) = esc" 1 ( ) = cot" 1 ( ). 

(c) Find tan (sin -1 f ), sin (tan -1 x). 

(d) Show that tan -1 \ + tan -1 \ = 45 . 



16. (a) Prove that sin (2 sin l x) = 2 x V 1 — x 2 . 



(b) sin' 1 1 + sin" 1 T 8 T + sin" 1 if = 



T 



2 

7. (a) If /(z)=^±^, find >'(*)• 

2T — I 

(&) Solve the equation, 

cos -1 x + cos -1 (1 — #) = cos -1 (— x). 



CHAPTER XIII 



TRIGONOMETRIC CURVES 

126. Functions Represented by Curves. The student is prob- 
ably already familiar with the fact that for every function of x,f (x), a 
curve or graph may be constructed which is said to represent that 
function. This curve is merely the totality of all the points whose 
coordinates x, y, satisfy the relation y = f(x). The actual construc- 
tion of the curve representing a given function y = f(x) consists 
in plotting a limited number of points, using for abscissas prop- 
erly chosen values of x, and for ordinates the corresponding values 
of y determined by the relation y = f (x). The smooth curve con- 
necting the points thus plotted is said to be the curve or graph rep- 
resenting the function y = / (x) . 

127. The Straight Line, y = mx + c. 

Suppose the given function is 

y = 2 x + i. 

We give x certain values and compute the corresponding values 
of y. Thus, if 

- i, - 2, - 3, - 4, - 5j etc -> 

- i> - 3i - S, ~ 7, ~ 9, etc - 
We now locate the point whose abscissa 
is 5 and whose ordinate is n, another 
point whose abscissa is 4 and whose 
ordinate is 9, and similarly each of 
the points (3, 7), (2, 5), (1, 3), (o, 1), 
(- 1, - i),etc. 

We then connect the points thus lo- 
cated by a smooth curve, in this case 
a straight line, Fig. 152. This line is 
said to be the curve or graph repre- 
Fig. 152. y = 2 x + 1. senting the function y = 2 x + 1, or, in 

short, the straight line y = 2 x + 1. 

253 



* = 5, 


4, 


3, 


2, 


1, 


0, 


y = ii, 


9, 


7, 


5, 


3, 


1, 













t 


r 




























C2, 


5) 




















fa 


3) 












































/o. 


1) 
































Y 





































-1, 


1) 




















(-4,-3 


) 










































h 


»,-5 


) 







































254 



PLANE TRIGONOMETRY 



[chap. XIII 



In the example just given, x may have one value as well as another, 
say 1,000,000 as well as 2 or 3, consequently the line representing the 
equation y — 2 x + 1 will be indefinite in length and we must con- 
tent ourselves with drawing only a portion of it. What portion this 
is to be depends on the purpose in view, but unless there is a special 
reason to the contrary, it is customary to construct the portion 
nearest the origin. 

In a similar manner every equation of the form 

y = mx + c, 

where m and c are known numbers, is represented by some straight 
line, m and c determining the direction and position of the line with 
respect to the coordinate axes. 

128. The Circle, x> 2 + y 2 = a 2 . 

Suppose the given equation is 



then 



x 2 + y 2 = 25, 



y = ±^25 — x 2 , 
and we have for corresponding values of x and y, 



x = 5, 
y = o, 
x = — 1, 
y = ± 4.89, 



4, 

±3, 

- 2, 
± 4.58, 



3, 

±4, 
- 3> 
±4, 



± 4.58, 
~ 4, 
±3, 



± 4.89, 

- 5. 
o. 



o, 

±5, 
etc., 

etc. 



If, as before, we construct the sep- 
arate points 

(5, o), (4, 3), (4, - 3), (3, 4), (3> - 4), 
(2, 4.58), (2, — 4.58), etc., and connect 
the points thus obtained by a smooth 
curve, we obtain the circle, Fig. 153, 
which is said to be the curve or graph 
representing the equation x 2 + y 2 = 25, 
or, in short, the circle x 2 + y 2 = 25. In 
this case x cannot be numerically greater 
than 5, for then y would be imaginary. 

In like manner every equation of the form 

x 2 + y 2 = a 2 , 

where a is some known number, is represented by a circle whose 
center is at the origin and whose radius is a. 













] 


r 






1 






















(3,4) 


























3-)- 






















V 












































(5 


0) 



























rar 














































/(A- 


=3)- 






















H-S 




















(3,- 


-4) 





























Fig. 153. x 2 +y 2 = 25. 



i3°] 



TRIGONOMETRIC CURVES 



255 



129. The Hyperbola, oc 2 — y 2 = a 2 . 

As another example, let us construct the curve whose equation is 



xr 



Solving for y, 



y 2 = 25. 



9> 

± 748, 
- 9, 
± 748, 



10 
± 8.66, 
etc. 
etc., 



y = dz \Zx* — 25. 

Corresponding values of x and y are 

* = 5> 6, 7, 8, 

y = o, ± 3.32, ± 4.90, ± 6.25, 

x =. - 5, - 6, - 7, - 8, 

y=+o, ± 3.32, ±4-9°. ±6.25, 

In this case x cannot be numerically less than 5, for otherwise y 
is imaginary. 

If we construct the separate points (5, o), (6, 3.32), (6, — 3.32), 
etc., and draw a smooth curve connecting them, we obtain the two 
curves PQ, P'Q r , Fig. 154. These curves constitute the two branches 
of a single curve, known as the hyperbola, more specifically as the 
equilateral hyperbola. 

It is easy to see from the equation that the larger x is, the more 
nearly will x and y be equal, that is, the branches of the equilateral 
hyperbola approach the straight lines 
PP r and Qff drawn through the origin 
and making angles of 45 ° with the two 
directions of the x-axis respectively. 

In like manner it will be found that 
every equation of the form 

x 2 — y 2 = a 2 , 

where a is some known number, is 
represented by an equilateral hyper- 
bola, a determines the distance from 
the origin at which the hyperbola 
crosses the x-axis. This is known as the semimajor axis of the 
hyperbola. 



W- J^ ^- 1£ 




X5 7 *Jr 


Stsaat: zzy 


^ s it dYhZ. 


v \ v£?*?»*»— 




T "^ v 7 




,rB,0)- J \7 -(570); 


" ~J&.± " £ 




~t / 5 A 


T 7 ~ X 




/ y (x-^pX 


A^dl. \\ 


_)• ' ^ 11 


<m ' ± \e> 


Bl ±5 



Fig. 154. x 2 - y 2 = 25. 



130. The Sine Curve, y = sin a?. 

Let us now construct the curve representing y = sin x. 
Since x may have any value either positive or negative, the curve 
representing the sine function will extend indefinitely in both direc- 



256 



PLANE TRIGONOMETRY 



[chap. XIII 



X = 



In radians, o, 



tions (right and left). Let us first construct that portion of the 
curve which corresponds to values of x between o and \ ir, that is, to 
angles in the first quadrant. Referring to the table of natural sines, 
we find the following corresponding values of x and y: 

In degrees, o°, io°, 20 , 30 , 40 , 50 , 6o°, 70 , 8o°, 90 . 

7T IT T 2 7T $ IT TV J TV \TV IT 

18 9 6 9 18 3 18 9 2 
y = o, 0.17, 0.34, 0.5, 0.64, 0.77, 0.87, 0.94, 0.98, I. 

In order to avoid awkward fractions, we will use ^ t as the unit 
along the #-axis.* 

With - for a unit, — = <t unit, - = i unit, etc., and we now readily 
3 18 9 

locate the points 

O = (o, o), P, = fe, 0.17), P 2 = fc 0.34Y . . . , P t = g, A 

Connecting these points by a smooth curve we obtain the curve 
OP1P2P3 . . . Ri (Fig. 155), which is the sine curve for the first 
quadrant. 



Y 












Bx 




















Fi j 














X 












A/ 










X 







IT 

6 

i 


J 
1 


r 
1 


2 



Fig. 155. y = sin x. 



We may now easily continue the sine curve through as many 
additional quadrants as we choose. 

Second quadrant. While x varies from J x to tt, sin x varies from 
1 to o. Moreover, since sines of supplementary angles are equal, 
ordinates equally distant from Ri will be equal, that is, the curve will 
be symmetrical with respect to the ordinate at Ri. Hence, con- 

* This will distort the curve slightly, since - = — — — = 1.0471+. 



i3i] 



TRIGONOMETRIC CURVES 



257 



turning from Ri, the curve will approach the x-axis, meeting it at 
R 2 (Fig. 156), whose distance from the origin is ir. 









\T 


R 1 












1 


1-2 












B 2 




T ~ 


—IT 


K 


IT 

2 




Ob 


7T 

2 


IT 




3 IT 
2 




2tt^ 






E-i 












R s 







Fig. 156. y = sin x. 

Third and fourth quadrants. While x varies from t to 2 ir, sin x 
is negative, the numerical value being the same as when x varies 
from o to ir. Hence, continuing from R 2 the curve will descend below 
the #-axis, reaching the lowest point at R3, where x = § x, and meeting 
the x-axis again at R4, where x = 2ir. The form of the portion of 
the curve below the x-axis will be like that above the #-axis when 
revolved about this axis through an angle of 180 . 

When x is increased or diminished by 2 ir, sin x has the same value 
as before, hence extending from i? 4 to the right or from to the left 
the curve repeats itself indefinitely, that is, the complete sine curve 
consists of an infinite number of waves or undulations of which 
OR 1 R 2 R^ is one. 

OR4, the distance between two consecutive points at which the 
curve crosses the x-axis in the same direction, is called the wave 
length of the curve. The greatest height of the curve, represented by 
the ordinate at Ri, is called the amplitude of the curve. 

A curve like the sine curve, which repeats itself at definite inter- 
vals, is called a periodic curve; the interval at which the repetition 
takes place is called the period. Likewise the function which such 
a curve represents is called a periodic function. 

The sine function is a periodic function whose period is 2 -k. 

131. The Tangent Curve, y = tan x. 

To construct the tangent curve for the first quadrant, we com- 
pute by means of a table of natural functions the corresponding 
values of x and y, as follows: 

30°, 



In degrees, o°, io°, 20 , 



x = i 



y = 



40 , 50 , 6o°, 7 o c 



8o°, oo°. 



In radians, o, — 5 
18 



7T 



2 7T 



IE, 
18 



T 



3 



7 7T 4_T T 
l8 9 2 



o, 0.18, 0.36, 0.58, 0.84, 1. 19, 1.73, 2.75, 5.67, 00 



258 



PLANE TRIGONOMETRY 



Ichap. xin 



Plotting the points obtained by using the x's for abscissas and the 
corresponding values of y for ordinates, and connecting these points 
by a smooth curve, we obtain the curve OR, Fig. 157, which rep- 
resents the equation y = tan x for values of x in the first quadrant. 



Since tan 



7T = 00 



the curve will not intersect the perpendicular at 



\ 7r, but will approach it indefinitely. 





1 


/£-! 




T R l 


f 






Ik 

1 
1 






B-2 


/ 


K\-\ 





J 


Bi 


Bi 




i 


&i -v 




r 


7T 




7T 7T 7T 

s |1 

1 




IT 


Jli7r 
2i 


/ 


2?r" 






■Bvi 1 






1 
1 






4» 1 


/ 





Fig. 157. y = tana;. 

Second quadrant. While x varies from \ -k to ir, the tangent varies 
from - 00 to o, hence between Ri and R-z the tangent curve will be 
below the x-axis, the numerical values of the ordinates being equal 
to those between and Ri taken in the reverse order. 

Tan (-7T + x) = tan x, hence beginning with R2 the curve repeats 
itself. The tangent curve, therefore, consists of an infinite number 
of disconnected branches. 

The tangent curve is a periodic curve, the tangent function is a 
periodic function, the period in each case is t. 

A line like R1R1 or R-iR-i, Fig. 157, to which the curve approaches 
indefinitely near without ever reaching it, is called an asymptote 
to the curve. The lines PP r and QQ ; , Fig. 154, are asymptotes 
to the hyperbola. An asymptote is a tangent to the curve at 
infinity. 

When the student has become familiar with the forms of the sine 
and tangent curves, he can readily sketch them from a very few points 
whose coordinates are known from memory. Thus, for the values 



7T 



X = o, -> 
6 



7T 

4 



T 



7T 
2 



the corresponding values of the functions are known without con- 
sulting a table, namely 

— = 0.71, — A 

2 



sin x = o, 0.5, 



= 0.71, — A = 0.87, 1, 



i 3 i] TRIGONOMETRIC CURVES 259 

tan x = o, — A = 0.58, 1, V3 = i-73> °° • 

The same remark applies to the sketching of each of the re- 
maining trigonometric f mictions. 



Exercise 57 

1. Construct the cosine curve. 

2. Construct the cosecant curve. 

(Suggestion. This curve is most readily sketched from the sine 

curve by remembering that y = esc x = — * ) 

sin # / 

3. Construct the secant curve, [y = sec x = )• 

\ cos x) 

4. Construct the cotangent curve, (y = cot x = ■)• 

\ tan x) 

5. Construct the curves whose equations are y = sin~ r ar,. 
y = tan -1 x. 

(Suggestion. If y = sin -1 x, then x = sin y, etc.) 

6. y = cos x = sin I- -\- x\- From this relation it follows that 

for every ordinate on the sine curve there is an equal ordinate on 
the cosine curve whose abscissa is the abscissa of the former dimin- 
ished by \ 7r, that is, for every point on the sine curve there is a point 
on the cosine curve, the latter being \ ir to the left of the former. Thus. 
we see that the cosine curve is merely the sine curve shifted a dis- 
tance J 7r to the left. By a similar reasoning show that the secant 
curve is the cosecant curve shifted a distance \ it to the left. 

7. y = cot x —— tan(- -f- x\. From this relation show that the 

cotangent curve may be obtained by shifting the tangent curve a 
distance of \ it to the left, and revolving it about the x-axis through 
an angle of 180 . 

8. y = cos x = sin(- — x\ = — sinfs: J. From this relation 

show that the cosine curve may be obtained by shifting the sine 



260 



PLANE TRIGONOMETRY 



[chap, xiii 



curve a distance of \ w to the right, and revolving it about the #-axis 
through an angle of 180 . 











2 


r 


1 
1 
1 














1 1 * 

/ ' * 
/ 1 l 

/ ■ \ 

1 1 V 




ft 


/ 1 
/ 1 

i 1 

7 i 


ol 










\ 1 
\ 1 






V 

■Or 


l 
l 


°/ 


\l 






N& 


7 


b\4 


V^ 


A/C\ 


i 


N& 


f s 


II 
\ ' 
\ ■ 


-iK 






J 2\ 


\/Y 





i 


/TV C 


■ 


! i 


i * 

7 \ 
t \ 






J * 
A. \ 

Y 




i 

i 
i 




N V 

/ \ 




1/ 






* i / 

* ! / 

* / 
1 ' 






< / 
i / 
i / 
i / 


\ 

\ 
1 
\ 




HP 1 

i i 
\ i 














j 









Fig. 158. The Six Trigonometric Curves. 

132. The Sinusoidal or Simple Harmonic Curves, 

y = a sin (Tex -f- €). 

(a) y = a sin x, a being constant. Each ordinate of the curve 
representing this equation is a times the corresponding ordinate of 
the curve y = sin x. The required curve is the curve obtained by 
lengthening or shortening (according as a is greater or less than 



] 


- 


a=2 
















1 
a4=i 
















1 

1 








2i 


r X 







7T 

2 


17 




i)7T 
















1 
1 

- j _y 























Fig. 159. y = a sin x. 

unity) the sine curve in the direction of the y-axis, leaving the wave 
length unchanged. The curve is a sinelike curve (sinusoid) whose 
amplitude is a. Fig. 159 shows three sinusoids of equal wave 
lengths and amplitudes J, 1, 2 respectively. 



1 



132] 



TRIGONOMETRIC CURVES 



261 



(b) y — a sin kx, a and k being constants. 

The curve representing this equation has the same amplitude as 
the curve y = a sin x, but the wave length differs, for it crosses 
the x-axis when 

kx = o, it, 2 ir, etc., 



that is, when 



X — o, 



7T 2 7T , 

-j — , etc. 

k k ' 



The distance between two consecutive crossings of the x-axis in 
the same direction is 2 ir/k, that is, the required curve is a sinusoid 
whose wave length is 2 ir/k. If we denote this wave length by X, we 
have 

X = — , from which k = — > 
k X 

and the equation y = a sin kx may be written 

y = a sin , X being the wave length. 



Y 


























V 























2 \ 


717 


^ 


I^N 


>y< 


27T 


ix\ 




37TX 

























Fig. 160. y = a sin (2 irx/\). 

Fig. 160 shows three sinusoids of equal amplitudes and wave 
lengths 7r, 2 7r, 3 7r respectively. 

(c) y = a sin (kx -\- e), a, k and e being constants. 

The curve representing this equation crosses the x-axis where 

kx + € = O, 7T, 2 7T, etc., 



that is, where 



x = j 



7T 



2 7T 



£ 



, etc. 



The ordinates of the highest and lowest points are a and — a 
respectively, and the distance between two points where the curve 
crosses the x-axis in the same direction is 2 ir/k. When x = o r 
y = a sin e. The curve is a sinusoid, amplitude a, wave length 2 ir/k, 



262 



PLANE TRIGONOMETRY 



[chap. XIII 



but instead of passing through the origin, it crosses the ^-axis at a 
distance a sin e above the origin. Fig. 161 shows three sinusoids of 



] 


T 


















W\ 






V) 


HI 

5 T 







7T\_ 

2fc N > 


S7T\ TC 




/iTL 


/fir/ 


27T ^* 

k 



















Fig. 161. 

equal amplitudes and wave lengths and e = o, J x and J 7r respectively. 
The third of these curves has for its equation 



y = a sin ( kx + - ) = a cos &#, 



which shows that every cosine curve is a sinusoid. 

Sinusoids are extensively used in physics to represent the motion 
of vibrating strings, tuning forks, and other vibrating bodies emit- 
ing musical sounds. For this reason they are often referred to as 
harmonic, or, more strictly, as simple harmonic curves. 



133 » Angles as Functions of Time. In many physical problems 
which give rise to equations of the form y = a sin (kx + e), the 
independent variable represents not an angle, but the time of an 
action or motion. 



A 



B -r> .3 


Y 




















* ^ 










in 

a 


* 


f 






< 


\ ° 







\ 


r— ex 






/ e 




B' 

















Fig. 162. 

Let 7, Fig. 162, be the initial position of a point moving in the 
^circumference of a circle (radius OP = a) with a constant velocity. 
Let P be the position occupied by the moving point, / seconds after 
leaving I, and let 00 represent the angular velocity of P, that is, the 
angle described by OP in one second of time. Then angle IOP = cot, 



133] TRIGONOMETRIC CURVES 263 

and angle A OP = wt + e, where e is the angle which 01 makes with 
some fixed diameter as AOA' . Furthermore, let M represent the 
projection of P on BB', the diameter perpendicular to AA', then 
OM = a sin (cot + e). The curve 

y = a sin (cot + e) (1) 

obtained by using equal distances on the axis of abscissas to repre- 
sent equal intervals of time, and using for ordinates the distances 
OM , which correspond to various values of t, enables us to see at a 
glance the position of M at any given time t. 

If T represents the time required by P to complete one revolu- 
tion we have 

co T = 2 7T, from which co = — > 

T 

so that (1) may also be written 

t/ = ttsinp^+€j, (2) 

where T is the periodic time, or period of oscillation of P and M. 

Since T represents the time required to complete one revolution, 
its reciprocal i/T will represent the number of revolutions completed 
in a unit of time. This value i/T is known as the frequency of the 
oscillation. If the frequency is denoted by v, equation (2) assumes 
the form 

y = asm (2 VKt-\- e). (3) 

Motion like that of the point M in Fig. 162, that is, any motion 
which can be expressed by an equation of the form y = a sin (bt + c) 
is called simple harmonic. The motion of vibrating tuning forks, of 
water waves, of an oscillating pendulum, of a galvanometer needle, 
of alternating currents, of sound and light and magnetic electric 
waves, are familiar examples of simple harmonic motions. 

Exercise 58 

Plot the following curves: 

1. y = sin 2 x. 2. y = sin- • 3. 3; = 3 sin-- 

2 3 

4. v = J sin 3 £. 5. y = sinf x + -)• 6. y = 2 sinf^— x— -J. 



264 



PLANE TRIGONOMETRY 



[chap. XIII 



7. Construct sinusoids having the following amplitudes and 
wave lengths: 

(a) Amplitude =1.5, wave length = 3.5. 

(b) Amplitude = 0.25, wave length = -• 

2 

(c) Amplitude = 1, wave length = Sir. 
Write the equation for each of these curves. 

Ans. (a) y = 1.5 sin 



4JTX 

7 



X 



(b) 4 y = sin 4 x, (c) y = sin - 

4 



8. Show that the equation 

y — a cos (bt + c) 
may be used to represent harmonic motion as well as the equation 

y = a sin (bt + c). 
9. Plot the curves (a) y = sin 2 x, (b) y = tan 2 x. 

10. Plot the curves (a) y 2 = sin x, (b) y 2 = tan x. 

11. A point moves in the circumference of a circle whose radius 
is 8.5, from an initial position whose angular distance from the 
right-hand extremity of a horizontal diameter is 15 , with a uniform 
velocity such as to complete a single revolution in 54 seconds. 
Write the equation between y and t, where y represents the vertical 
distance of the point from the horizontal diameter at any given 
time t. 

12. A piece of paper is wrapped around a 
wooden cylinder and then an oblique section is 
made by sawing the cylinder in two. If the 
paper is now unrolled and laid flat, its edge will 
form a sinusoidal curve. Prove it. 



Ans. The equation of the curve is y = a sin - > 



where PR = y, OR = x, AT = a, and OC = r = 
the radius of the cylinder. 



:l_y- 




p 










> 



Fig. 163. 



134. Composition of Sinusoidal Curves. Curves representing an 
equation of the form 

y = a-i sin (b l ac + c t ) + a 2 sin (b 2 oc-\-c 2 )+ etc., 
may be readily constructed from the component curves 

Vi = #i sin (bi x + ci), yi = (h sin (b 2 x + c 2 ), etc., 



134] 



TRIGONOMETRIC CURVES 



265 



for since 

y = y\ + yi 

the ordinate of any point on the required curve is found by adding 
the corresponding ordinates of the component curves. 

Example i. Plot the curve y = sin x + cos x. 
Solution. Plot separately the two curves 

y\ = sin#, [curve (1), Fig. 164], 

y 2 = cos x, [curve (2), Fig. 164]. 















F 


I 


p 














>< 

/ 

/ 


s <? 






t 

/ 

s 








v s 


N 




s 










\ 
\ 


x\ 


\ 


/ 
/ J 


s 

s 





f\ 


$ 


G 


V 


s / 


V 

• 


X 




















v 2 











Fig. 164. 

The required curve y = ?i + y 2 [curve (3), Fig. 164] is then 
obtained by adding the two ordinates corresponding to any given 
value of x. Thus, if x = OF, y = FP = FP X + FP 2 . It is im- 
portant to remember that for points below the x-axis the ordinates 
are negative; thus, if x = OG, y = GQ = GQi + GQ 2 . Now GQi is 
positive but GQ 2 is negative. GQ 2 being the longer of the two, their 
algebraic sum, that is, GQ, will be negative. 

It should be noticed that certain points, as, for instance, those for 
which y = ± 1, ± 2, or o, may be located at sight. After the shape 
of the curve is known, these points suffice to sketch the required 
curve. 

Each of the sine curves Vi = sin x and y 2 = cos x has the period 
2 7r and the resultant curve y = sin x + cos x is another sine curve 
with the period 2 jr. This may be shown analytically as follows: 

V 



y = sin x + cos x — sin x + sin I # ]> 

= 2 sin - cos f - — A by formulas in Article 113, 
4 \4 / 

= V2 cos (- — x] = V2 sin (x J r 7r 

u / V 



266 



PLANE TRIGONOMETRY 



[chap, xm 



that is the curve y = sin x + cos x is a sinusoid having an ampli- 
tude V 2, period 2 7T, and crossing the x-axis at the point x=-\ ir . 

The method employed in constructing the curve in the preceding 
example applies equally well to the compounding of any number of 
curves. The ordinates of the resultant curve are always the alge- 
braic sums of the ordinates of the component curves. 

Example 2. Plot the curve y = sin x + sin 2 x. 

Solution. Construct separately the two curves 

yx = sinx, [curve (1)], 

y 2 = sin 2 x, [curve (2)], 
then 

y = y\ + 3>2 = sin x + sin 2 x 
yields the curve (3), Fig. 165. This curve has the period 2 w. 



Y 


(?) 




















1 / 
1/ A 


* 


\ 
\ 
\ 

V 






/ 
if \ 


\ 
\ 











\ 
\ 
\ 


/ 

/ 


N 


\ \ 

\ \ 


A 


X 



















Fig. 165. 

Example 3. Plot the curve y = sin x -f J sin 3 3 + J sin 5 x. 
Solution. 

yi= sinx gives curve (1). 
y 2 = i sin 3 x gives curve (2), 
3^3 = i sin 5 x gives curve (3), 
? = yi + yi + y3 gives curve (4). 
This curve also has the period 2 w. 



Y 




Cl) 














rCx 




\ 




«-'"~> 









J, *._ 


(2) 








<1 


X 



















Fig. 166. 



135] TRIGONOMETRIC CURVES 267 

135. Theorem. The resultant • of two simple harmonic curves 
having equal wave lengths is another simple harmonic curve having the 
same wave length. 

Proof. Let the given curves be 

vi = aisinl— x-\- cij, y 2 = ^sinf — x + c 2 ), 

X being the common wave length. (Art. 132.) 

•Now 

• /2 7T \ - /2 7T \ • 

Vi = ai sin f — x 1 cos ci + Oi cos ( — x I sin d, 

y<i = 02 sin ( — - x \ cos c 2 + 02 cos ( — x J sin c 2 , 
therefore 

y = yi + ^2 

= (ai cos ci + 02 cos C2) sin f ~ x J + (ai sin c\ + #2 sin c 2 ) cos [ — # J 
= a cos c sin ( — x ) + a sin c cos f — x ) 

= asint—x-\- c), (j) 

where 

a cos c = a\ cos c\-\- 02 cos c 2 , 

, • (2) 

a sin c = #i sin Ci + 02 sin c 2 . 

Dividing the second of the equations (2) by the first, gives 

tan C = a '- sinCl + a2sinC2 , (3) 

aiCOSCi+ 02COSC 2 

and taking the sum of their squares 

a 2 (cos 2 c + sin 2 c) = Oi 2 (cos 2 Ci + sin 2 C\) + 2 0\02 (cos Ci cos £ 2 
+ sin C\ sin c 2 ) + O2 2 (cos 2 c 2 + sin 2 c 2 ), 

or a 2 = ai 2 + a 2 2 + 2 fli^ cos (ci — c 2 ). (4) 

Equation (1) shows that the resultant curve is a simple harmonic 
curve whose wave length is X. The amplitude a of the resultant 
curve is given by (4), and the constant c by (3). 



268 PLANE TRIGONOMETRY [chap, xnr 

136. Fourier's Theorem. Before leaving the subject of har- 
monic curves, we will state in simple language and without proof a 
most famous theorem, which in the language of Thomson and Tait * 
" is not only one of the most beautiful results of modern analysis, 
but may be said to furnish an indispensable instrument in the treat- 
ment of nearly every recondite question in modern physics. To 
mention only sonorous vibrations, the propagation of electric signals 
along telegraph wires, and the conduction of heat by the earth's 
crust, as subjects in their generality intractable without it, is to 
give but a feeble idea of its importance." 

Any arbitrary periodic curve can be considered the resultant of a 
sum of simple harmonic curves, and can therefore be expressed by an 
equation of the form 

y = ai sin (b\X + C\) + a^ sin (b 2 x + c 2 ) + etc. 

The importance of the theorem lies in this, that every periodic 
phenomenon whose changes can be measured can be represented by 
a periodic curve. Fourier's theorem shows how every such phe- 
nomenon can be resolved into a series of simple harmonic motions. 

Exercise 59 

1 . Plot the curve y = sin x — cos x. 

Show analytically that this curve is a sine curve, whose amplitude 
is V 2, and which crosses the #-axis at the distance \ it to the right of 
the origin. 

2. Construct the resultant curve of which 

yi = coslx -\-~), and y% = cos( x — - \, 

are the components, and show that the equation of the resultant 
curve may be written y = cos x. 

Construct the following curves: 

3. y = sinx + § sin 3 x. 4. y = sin 2 x -f- sin 3 x. 
5. y = sin x + J sin 4 x. 6. y = 2 sin x — sin 2 x. 

7. y = cos 2 x + 4Cos#. 8. y = sin# — J sin 3 x-\- % sin 53;. 

9. x = tan -1 (1 + \/y) + tan -1 (1 — V;y). 

* Elements of Natural Philosophy, Second Edition, Chapter I. 



137] TRIGONOMETRIC CURVES 269 

10. Plot the curve resulting from compounding the two curves 

vi = 2 sin (x + jjj, y 2 = 3 sin (x - -\ 

and show that the equation of the resultant curve is 
y = a sin (x-\- c), 

where a = V 19, c = tan -1 ( — — 3 J. 

11. Show that 

asin<£ -f- bcos<j) = V a 2 + 6 2 sin(0 + tan -1 -)- 

12. The force (inertia force) on the piston of an engine is given 
by the equation 

F = F (cos6 + jcos2 e\ 

where 6 is the angle which the crank arm makes with a fixed direc- 
tion, R the length of the crank arm, L the length of the connecting 
rod, and F Q a constant (the ideal centrifugal force at the crank pin 
center). Plot the curve, showing the value of F for any given posi- 
tion of the crank, when F = 600, R = 15, L = 45. 

13. s, the distance of the piston of a steam engine from its ex- 
treme position corresponding to 6 = o, is given approximately by 
the equation 

s — R 1 1 — cos 6 H sin 2 6 ), 

\ 2L J 

where R, L and 6 have the same meaning as in Example 12. Plot the 
curve showing the relation between s and 6, when R = 10 and L = 20. 

137. The Logarithmic Curve, y = log 10 as. 

With the aid of a table of common logarithms and a knowledge 
of the fundamental properties of logarithms, we find the following 
corresponding values of x and y: 

10, 100, ., ., 00 . 

O, I, 2, ., ., CO. 

O.I, O.OI, ., ., o. 

0,-1, -2, ., ., -00. 

In any case, if x = - , y — — log n. 
n 



Jv ■""" X • ^« 


3, 


4, 


5, 


y = 0, 0.30, 


0.48, 


0.60, 


0: 


X = 2) 


1 
3? 


1 

4) 


1 

5? 


y = - 0.30, 


- 0.48, - 


- O.60, " 


- 0. 



270 



PLANE TRIGONOMETRY 



[chap. XIII 



Y 



























( 


(1,0) 






X 



























These values enable us to plot the portion of the curve shown in 
Fig. 167. They also give us a definite idea of the shape of the curve 
beyond the limits of the figure. To the 
right the curve diverges more and more 
from the #-axis as x increases, below 
the #-axis the curve approaches the 
y-axis more and more nearly as x ap- 
proaches o; that is, the logarithmic 
curve y = logi x has the y-axis for an 
asymptote. 

Fig. 167. y — logio*. 

138. The Exponential Curve, y = io* 5 . 

Taking the common logarithm of each 
side of the equation y = io x , we obtain 

logio y = x, or x = log™ y. 

This shows that the exponential curve 
y = io* may be obtained from the curve 
y = logio# by interchanging the x's and 
y's, that is, by using the ordinates of 
the logarithmic curve for abscissas and 
lg ' z ' y ' the abscissas for ordinates. The result- 

ing curve is shown in Fig. 168. 

139. The Exponential Curves, y = e hx , where k is any constant, 
and e = 2.7 18+ , the base of the natural system of logarithms. 

(a) y = e x . Let us first consider the special case for k = 1. 
A general idea of the shape of the curve y = e x may be gathered 
from the following sets of corresponding values of x and y: 









Y 


























I 












(o,D 















X 



1 

2) 



X — O, 2> J > 2 ) 

y = i, (2.7)* = i.6, 2.7, (2.7)2 = 7.4, 



— i 



x = — 



2) 



— I, 



y = ^— = 0.61, — = 0.38, 
1.6 2.7 



- 2 3 

I 



7-39 



= 0.14, 



oo, 
00, 

— 00. 

— o. 



From these values the curve may be sketched as in Fig. 168. 

If the curve is to be plotted with greater accuracy than the 
above figures will permit, it is best to employ logarithms. For, on 



139] TRIGONOMETRIC CURVES 27 1 

taking the logarithm of each member of the equation y = e x } we 
have 

logio y = x logio e = x logio 2.718+ = 0.4343 x. 

Assigning to x in succession the values o, 0.1, 0.2, ., ., 1, 2, 3, etc. 
we obtain by means of a table of logarithms, 

x = o, 0.1, 0.2, ., ., 1, 2, 3, etc., 

log;y = o, 0.04343, 0.08686, ., ., 0.4343, 0.8686, 1.3029, etc., 
y = 1, 1. 1052, 1. 2214, ., ., 2.718, 7.390, 20.085, etc. 

The intervals for x may be chosen as small as we please, and the 
table extended at will, depending on the accuracy desired and the 
extent of the region for which the curve is to be plotted. The 
values of y corresponding to negative values of x are the reciprocals 
of the values of y when x is positive. 

(b) y = e kx , k positive. 

When x = o, y = 1, therefore, no matter what value k has, the 
curve passes through the point (0,1) Fig. 169. 

Let k > 1. So long as x is positive, e x > 1, and therefore e kx = 
(e x ) k > e x , that is, to the right of the y-axis the curve y = e kx , k > 1, 
lies above the curve y = e x , and it will diverge from it the more, the 
greater the value of k. For negative values of x, k > i,e kx = (e x ) k <e x , 
that is, to the left of the y-axis the curve y = e kx , k > 1, will lie 
between the curve y = e x and the x-axis and will converge the more 
rapidly to this axis the greater the value of k. For any given value 
of k the curve may be roughly sketched by means of a few properly 
chosen points. If greater accuracy is required, we first compute a 
table of values from the relation logio y = 0.4343 kx. 

Considering in like manner the cases when k < 1, we find that the 
curves y = e kx , k < 1, lie between the curve y = e x and the straight 
line drawn parallel to the x-axis through the point (o, 1) and that the 
curve will approach this straight line more nearly the smaller the 
value of k. Fig. 169 shows the curves for the values k = \, k = J, 
k = 1, k = 2 and k = 3. 

(c) y = ^ x , k negative, or y = e~ kx , k positive. 

The curve y = e~ kx (1) is most readily obtained from the curve 
y = e** ( 2 ). For y = e ~ kx = (e k )~ x , from which it is plain that for 
a positive value of x, the ordinate y of (1) will be the same as the 



272 



PLANE TRIGONOMETRY 



[chap. XIII 



value of y in (2) for the corresponding negative value of x, and vice 
versa, the y in (1) when x is negative will be equal to the y in (2) 
when x is positive. This means that the two curves y = e~ kx and 
y = e kx , k being the same in both cases, are symmetrical with re- 
spect to the y-axis, so that either one being given, the other may be 
traced from it without computing anew the coordinates of its points. 
Two such curves have the same relation to each other as an object 
and its reflection in a mirror. For this reason either of the two is 
said to be the reflection on the y-axis of the other. Fig. 169 shows 
the curves y = e kx for the values k = — 3, — 2, — .1, — §, — J. 




Fig. 169. y = e kx . 



140. The Compound Interest Law. In physics, chemistry and 
various branches of engineering, related quantities occur which are 
subject to laws which may be expressed by the formula 

y = ae kx , (1) 

or by the equivalent formula 



x — - log-' 
k a 



(2) 



In fact, it can be shown that whenever two quantities are so related, 
that the ratio between their changes is always proportional to one 
of the quantities, the relation between them may be expressed by 
either (1) or (2), where x and y are the quantities in question, and k 
and a constants which depend on the rate of change and the initial 
values of x and y. The amount of money due at any time on a sum 
of money put out at compound interest, the interest being added to 
the principal not at stated intervals but as fast as it accrues, varies 
with the time according to this law. For this reason the general law 



i4o] TRIGONOMETRIC CURVES 273 

expressed by (1) or (2) is commonly known as the compound interest 
law. The student of science will meet numerous examples of the 
compound interest law. A few simple examples are given in the set 
of problems which follows. 











Exercise 


60 


Plot the 


following 


curves 




X 




1. 


y = 


log e X. 


2. 


y = 


IO^. 




4- 


y = 


X 

io" 1 . 


5- 


y = 


Jloge 


X 

2 


7. 


y = 


3 X - 


8. 


y = 


3" x . 




10. 


y = 


2~ 2X . 


11. 


y = 


2 e x 





3. y = io" x . 

X 

6. y = 3 • io 3 . 
9. y — 2 2x . 
12. y = — 2 e x . 

13. Given the curve y = e kx , trace the curve y = — e kx without 
computing the coordinates of its points. 

(Suggestion. The required curve is the reflection on the #-axis of 
the given curve.) 

• 14. The amount A due on a principal of P dollars, put out at 
compound interest at r per cent for t years, the interest being added 
to the principal as fast as it accrues, is given by the formula 

rt 

A = Pe m . 

Plot the curve showing the amount due at any given time, when 
P = 100 and r = 5. 

(Suggestion. Use different values of t for abscissas and the cor- 
responding values of A for ordinates.) 

15. The work W due to the expansion of steam in the cylinder of 

a steam engine, while expanding from a given volume to a volume V, 

is given by the formula 

W = a log e V - b, .. 

where a and b are two constants depending upon the initial volume 
and pressure. Plot the curve showing the relation between V and 
W for any volume from V = 1 to V = 4, when a = 15,000, and 
b = o. 

16. Newton's law of cooling. The difference between the tem- 
perature of a body and the temperature of the medium surrounding 

it is given by the formula 

6 = be'* 



274 



PLANE TRIGONOMETRY 



[chap. XIII 



where a and b are constants depending upon the nature of the body 
and the initial temperatures. If when t = o, 6 = 50 , and when 
t = 60 seconds, 6 = 25 , plot a curve showing the temperature 6 at 
any time / up to 60 seconds. 

c I x - -\ 
141. The Catenary, y = _( e « _j_ e c J in which e is the base of 



the natural system of logarithms and c any positive constant. 

We first consider the special case c = 1, y = \ (e x + e~ x ). 

Put y\ — e x and y 2 = e~ x , then y = | (3/1 + y 2 ), which shows 
that any ordinate of the required curve is equal to half the sum of 
the corresponding ordinates of two curves Vi = e x and y 2 = e~ x . 
If, therefore, these two curves are drawn first, as in Fig. 170, the 









■ Y 




• if 
1 I 
1 l[ 

! /v 








\ \ 


\ 
\ 


1 

,f 


k./ // 


/c\> 










\\ 


/ 
/ / 












„- 


/' 


N ^ v j 


s 















A 






J£ 



Fig. 170. The Catenary. 

required curve is easily drawn by taking each ordinate, as AP, equal 
to half the sum of the ordinates A Pi and AP 2 . 
It should be observed that 

2 2 2 

that is, each point P bisects the line joining two corresponding points 
Pi and P 2 . 



We now pass to the general case, y = - \e c + e c ) may be written 



(1) 
(2) 





X X 

y e c + e c 




c 2 


which becomes 


2 


where 


, y / # 
y = z » r = - 



142J 



TRIGONOMETRIC CURVES 



275 



From this we see that if the coordinates x', y' of any point on the 
curve (2) be multiplied by c, the resulting numbers are the coordi- 
nates x, y of some point on the curve (1). The curve (1) is therefore 
merely some magnification of the curve (2). The curve in Fig. 170 
may therefore be taken to represent any equation of the form (1), 
provided the proper scale is employed in the construction. It is 
only necessary to let each unit of length along the coordinate axis 
represent c units. 



The curve y = -\e c + e c ]is called the catenary. It is the curve 

formed by a rope or flexible cable suspended between two points. 
c is the ratio of the horizontal tension to the weight of a unit length 
of the rope or cable. 

142. The Curve of Damped Vibrations, y = ae~ kx sin (coc + d). 
Put 



y\ — ae kx , (1), y 2 = sin (ex + d), (2) 



then 



y = yi>'2 = ae kx sin (ex + d), (3) 

that is, any ordinate of the required curve is equal to the product of 
the corresponding ordinates of the two curves (1) and (2). 




Fig. 171. The Curve of Damped Vibrations. 

Let the curves (1) and (2) first be constructed separately, as in 
Fig. 171. Now observe: 

(a) Since y 2 is less or at most equal to 1, y of (3) is less or at most 
equal to yi, that is, the required curve lies below the curve (1). 
> (b) Whenever yi or y 2 equals zero, y of (3) equals zero also, hence 
the required curve crosses the #-axis at the points iVi, N 2 , N&, N^ 
etc., at which the curve (2) crosses this axis. 



276 PLANE TRIGONOMETRY [chap, xm 

(c) Whenever y 2 = 1, as at H h H 3 , etc., y of (3) equals y h hence 
the required curve touches the curve (1) (since it cannot cross it) 
at the points K h K3, etc. 

id) Whenever y 2 = — i,~as at H 2 , H h etc., y = — y lm This enables 
us to locate the points K 2 , K4, of the required curve. 

The shape of the curve (3) is now apparent. It is a wave curve of 
constant wave length, but the amplitude of the successive waves 
diminishes. The rapidity with which the amplitude decreases 
depends on the value of the constant k in (1). This constant k is 
known as the logarithmic decrement of the curve. 

Like the exponential curve and the sine curve, this curve finds 
frequent applications in science. While the sine curve represents 
free vibrations, that is, vibrations not retarded by friction or other- 
wise, the curve y = ae~ kx sin (ex + d) represents damped vibrations, 
that is, vibrations suffering resistance of some kind. A pendulum 
vibrating in air or water, waves propagated in a viscous fluid and oscil- 
latory discharges from an electric condenser, are familiar examples 
of damped vibratory motion. 

Exercise 61 
Plot the curves: 



n x 



1. v = e cos x. 2. y = e* sin x. 3. y = 



e x — e' 



4. y = . (Suggestion. The given equation may be 

e x — e~ x \ 

1 / e x — e~ x \ 

written y = — , where y = 

y 2 / 



y 

2 



5. y = . See suggestion under Problem 4. 

e x -\- e x 



6. y = e - ^~zr- (Suggestion. The given equation may be 



,-x 

e x + e~ x 
w 



v-i e x — e x e x ~\~ e x \ 

ritten y = —, where vi = , y 2 = - 

y 2 2 2 / 

j t y — — ' - . i (Suggestion. This equation is the reciprocal of 



e" — e 
that in Problem 6.) 

8. The six functions 



e x — e~ x e x + e~ x e x — e~ x 

y = , y 2 = , y s = _ , 

J 2 2 e x + e x 



142] 



TRIGONOMETRIC CURVES 



277 



y A = 



e* — e 



y* 



e x + e 



y* 



e x + e 



are known as the hyperbolic func- 
tions. Show that 



y^ — y 2 = 1. 



2 _ 




yb 2 + yi 

y& 2 — y* 2 = 1. 

Compare the graphs of these func- 
tions, Fig. 172. The graphs are 
numbered in correspondence with Fig- 172. The Six Hyperbolic Curves, 
the suffixes of the y's in their equations above. 

9. A cable, weighing one pound to the foot, is suspended between 
two piers under a tension of 100 pounds. Plot the curve which it 
forms. 

10. The displacement of the end of a spring, from its position oi 
equilibrium, at the time t seconds is given by the equation 

s = ae Kl sin — t, 

where a is the amplitude of the vibration of the string if there 
were no friction, k represents the effect of friction retarding the vi- 
bration, and T is the time of a single vibration. Plot a curve show- 
ing the displacement at any moment during the first ten seconds, if 



CHAPTER XIV 

TRIGONOMETRIC REPRESENTATION OF COMPLEX QUANTITIES 

143. Imaginary Numbers. If we solve the equation 

x 2 + i = o, 



.we obtain x = ± V ' — i. Similarly, every equation of the form 

x 2 + a 2 = o, 

where a is any real number, has for its solution 

x = ±V- a 2 = ± flV^T, 



which contains V — i as a fa ctor. 

This new number V — i, whose square is — i, is commonly 
denoted by the letter i and is called the imaginary unit. 

Since i 2 = — i, it follows that 

i 3 = — i, i 4 = i, i 5 = *', i 6 = — i, P = — i, i 8 = i, etc., 

that is, 

Every integral power of i is equal to i, i, — i, or — i, according as 
the exponent of the power when divided by 4 leaves the remainder o, i, 
■2, or j. 

In symbols 

{*"= i ? ^n+l = ^ j4n+ 2 = - I} ^n+3 = _ £ 

Numbers like ^, 22, — $i, -> va*i, etc., are called imaginary 

2 

numbers or quantities. Every imaginary number consists of the 

imaginary unit i multiplied by some real coefficient. From the rule 

for the powers of i it follows that every even power of an imaginary 

number is a real number, every odd power of an imaginary number 

is again an imaginary number. 

144. Geometrical Representation of Imaginary Numbers. 

Every positive or negative real number x may be represented 
geometrically by a distance on the x axis measured to the right or 
left according as x is positive or negative. Equal distances measured 

278 



*44] 



COMPLEX QUANTITIES 



279 



in the same direction represent equal numbers, but equal distances 
measured in opposite directions represent numbers which are equal 
in magnitude but opposite in sign. Thus, in Fig. 173, if the points 
X-2, X-i, O, Xi, X 2 are taken at equal intervals, and OXi represents 
unity, then each of the segments 

OXi, XiX 2 , X-2X-1, X-iO represents +1, 

and each of the segments 

OX-i, X-1X-2, X 2 Xi, XiO represents — 1. 

In general, if the segment PQ represents the number a, QP repre- 
sents the number — a, and therefore PQ + QP — a •+ (— 0) = o. It 
follows that a line segment PQ on the x-axis will continue to repre- 
sent the same real number if it is moved along 4 the x-axis from one 
position to another, so long as its direction remains unchanged. 

If all the line segments have the same initial point O, as OXi, OX 2 , 
OX_i, OX-2, Fig. 173, the terminal points Xi, X 2 , X-i, X_ 2 will 
represent the numbers quite as well as the segments themselves. 
Thus, if OXi= 1, the numbers 1, 2, — 1, — 2 are represented equally 
well by the segments OXi, OX 2 , OX-i, OX-2, and by the points 
Xi, X 2 , X-i, X-2, respectively. 

Likewise every positive or negative imaginary number iy may be 
represented geometrically by a distance on the y-axis measured 
upwards or downwards according as y is positive 
or negative. Equal distances on the y-axis meas- ^ 
ured in the same direction represent equal imagi- 
nary numbers, equal distances on the y-axis 
measured in opposite directions represent imagi-/ 
nary numbers which are equal but opposite in sign. 
Thus, in Fig. 173, if the points F_ 2 , F_i, 0, Y h F 2 
are taken at equal intervals, and OYi is taken in 
length equal to 0X h then each of the segments 
OYi, FiF 2 , F-2F-1, F_iO represents + '*, and 
each of the segments 0Y- h F_iF_ 2 , F 2 Fi, Y& represents - i. 
In general, if the segment RS represents the number ai, SR repre- 
sents the number — ai, and therefore RS + SR = ai-\- {—ai) = o. 
It follows that any line segment on the y-axis will continue to repre- 
sent the same imaginary number if it is moved along the y-axis from 
one position to another so long as its direction remains unchanged. 



p 




xs 


Pi 












y 2 / 








y\ 




X- 2 


XV / 


K) x y 

fa/ 

A 


X 2 


■«i 




Y-i 


P 4 



Fig. 173- 



280 PLANE TRIGONOMETRY [chap, xiv 

If all the line segments have the same initial point 0, as OFi, 
OY 2 , OY-i, OY-2, the terminal points Y h F 2 , F_i, F_ 2 will represent 
the imaginary numbers quite as well as the segments themselves. 
Thus, if OFi represents i, the numbers i, 2 i, — i, — 2 i are equally 
well represented by the segments OY h OY 2 , OY- h OY- 2 , and by the 
points Fi, F 2 , Y-i, F_ 2 , respectively. 

When the points on the x-axis and v-axis are used to represent 
real and imaginary numbers respectively, these axes are referred to 
as the axis of reals and the axis of imaginaries respectively. 

The reason for choosing the axis of imaginaries at right angles to 
the axis of reals is found in the following considerations: 
The equation i 2 = — 1 may be put in the following form, 

+ 1 : i = i : — 1, 

that is, i is a mean proportional between +1 and — 1. Now the 
geometric construction for a mean proportional gives the perpen- 
dicular OFi, erected at O, to the semicircle constructed on X-1X1 
a? a diameter. Hence, if OXi represents +1, and OX-i represents 

— 1, OFi will represent V + 1 X — 1 or i. 

Or, we may reason as follows: i 2 = i X i = — 1, that is, two 
successive multiplications by i have the same effect as multiplication 
by —1. Now multiplying any line segment OXi by — 1 gives 

— OXi = X\0 = OX -1, that is, multiplying by — 1 has the effect 
of turning OXi through an angle of 180 , hence multiplying by i 
should have the effect of turning OX\ through half this angle or 90 , 
that is, i X OX\ = OFi, so that if OXi represents 1, OFi must 
represent i. 

145. Geometrical Representation of Complex Numbers. 

If we solve the equation x 2 — 4^+13 = 0, 
we obtain #=2±3'V— i } 

that is, x = 2 + 3 i, or 2 — 3 i. 

Each of these numbers consists of two parts, a real part + 2, and an 
imaginary part + 3 i or — 3 i. A number like 2 + 3 i, or 2 — 3 i, 
which consists of two parts, one which is real and the other imaginary, 
is called a complex number. The general form of a complex number 
is a + bi, where a and b are real numbers, a and b may be positive 
or negative, integral or fractional, rational or irrational. 



146] COMPLEX QUANTITIES 281 

A complex number a -f- bi is represented geometrically by a line 
segment joining the origin O to the point whose coordinates are a 
and b. Thus, the complex number 2 + 3 i is represented by the line 
segment OP\, Fig. 173, obtained by joining the origin to the point 
Pi, whose coordinates are 2, 3. Here also the direction of the line 
segment as well as its length is to be considered. The line-segments 
OPi, OP2, OP3, OP4 are equal in length, but they do not represent 
the same numbers. OP\ represents the number 2 + 3 i, OP 2 rep- 
resents — 2 + 3 i, OPz represents — 2 — 3 i, and OP 4 represents 
2 - si. 

Two directed line segments which have the same length and the 
same direction are considered equal, and may be taken to represent 
the same number. It follows that a line segment will continue to 
represent the same number if it is moved parallel to itself. Thus, 
OPi, X-2F3, P 3 0, F-3X2 being parallel and equal in length, all rep- 
resent the same complex number 2 + 3 i. 

If the directed line segments all have the same initial point O, as 
OPi, OP 2 , OP 3 , OP 4 , the terminal points P h P 2 , P3, P4 may be taken 
to represent the complex numbers as well as the directed segments. 
In this sense we may speak of the points Pi, P 2 , P3, P4 as represent- 
ing the numbers 2 + 3 i, — 2 + 3 i, — 2 — 3 i, 2 — 3 i, respectively. 

If in a + bi we put b = o, we get all possible real numbers a ; if 
a = o, we get all possible imaginary numbers bi, so that the complex 
numbers a + bi include as special cases all real and imaginary num- 
bers. 

146. Trigonometric Representation of Complex Numbers. 

y Let P (Fig. 1 74) represent the point x + iy. 

t/X, Let r denote the length OP and 6 the angle which 
OP makes with the #-axis. Then 

a? = rcos8, y = rsindy (1) 

Fig. 174. and 

ac -\- iy = r (cos + i sin 0). (2) 

r (cos 6 + i sin 6) is called the trigonometric form of the complex 
number x-\-iy; r, which is always positive, being the distance of 
the point P from the origin 6, is called the modulus or absolute value, 
and 6 is called the argument or amplitude of the complex number 
x + iy. 




282 



PLANE TRIGONOMETRY 



[chap. XIV 



From (i) we obtain 



cos0 = -, sin0 = t 7 , tan9 = ^, r = Va* + y 2 . (3) 

V V 3C 

The equations (3) enable us to express any complex number in 
the trigonometric form. Thus, if x + iy = 3 + 4 i, we find 

sin0 = f, tan0 = f, 



so that 



r = v 3 2 -j- 4 2 = 5, cos = 3 



5; 



3 + 4^5(cos53 8 , + zsin53 8 / ). 

If x -f- iv = 3 — 4 2', # = 3, y = — 4, and we have 

r = V 3 2+(- 4 )2 = 5, cosfl = I, sin0 = - f, tan0 = -f, 

from which 6 = 306 52' or — 53 8 r , 

so that. 3 — 4 ^* = 5 (cos 306 52' + i sin 306 52'), 

or 3-4*'= 5 (cos- 53°8 / + isin- 53°8'), 

= 5(cos53 8 , -isin 5 3 8 / ). 

Conversely, if a complex number is given in the trigonometric 
form, equations (1) enable us to express it in the form x + iy. 
Thus, if the given number is 2 (cos 30 + i sin 30 ), r = 2, 6 = 30 , 
and we have from (1) 

x = 2 cos 30 = V 3, v = 2 sin 30 = 1, 
so that 

2 (cos 30 + £ sin 30 ) = v 3 + *. 

147. Geometric Addition and Subtraction of Complex Numbers. 

Let OP and OP' (Fig. 175) represent 
any two complex numbers x + iy and 
x' + iy' respectively. Complete the par- 
allelogram OPQP', having OP and OP' for 
two of its sides. Draw QC perpendicular 
and PD parallel to OX. The right tri- 
angles PDQ and OBP' are equal (Why?), 
hence 

PD== OB = x', - DQ = BP' = y', 

and therefore 

OC = OA + PD = x + x', CQ = 4P + DQ = y + /, 
so that the directed line OQ represents the complex number 

(x 4- x') + i(y + y') = (x + *y) + W + */). 




148] COMPLEX QUANTITIES 283 

The sum of two complex numbers represented by OP and OP' re- 
spectively is represented by the diagonal drawn from of the parallelo- 
gram formed with OP and OP' as sides; or, 

To find geometrically the sum of two complex numbers represented by 
OP and OP' respectively, move OP' parallel to itself so that its initial 
point falls on the terminal point P of OP. The directed line, drawn to 
connect the initial point O of OP to the terminal point of OP' in its new 
position, represents the required sum. 

In its second form this rule may be easily extended to the sum of 
any number of complex numbers. To add geometrically the num- 
bers represented by OP, OP\, OP2, OP%, etc., we first add any two 
of them, as OP and OP\, to their sum we add any third, as OP 2 ; 
to the sum of these three we add a fourth, as OP 3 , etc. Leaving out 
the lines which are not needed to obtain the final result, we obtain 
the following rule: 

Construct a broken fine OPQRS . . . such that OP coincides 
with OP, PQ is equal and parallel to OP\, QR is equal and parallel 
to OP 2 , RS is equal and parallel to OP 3 , etc. The directed line 
joining O to the terminal point of this broken line represents the 
required sum. 

The length of the line representing the sum of two or more com- 
plex numbers will be less, or at most equal, to the sum of the lengths 
of the several lines representing the numbers added, hence 

The modulus of the sum of any number of complex numbers is less 
than, or at most equal to, the sum of their moduli. 

Geometric subtraction follows from geometric addition. Let OQ 
and OP (Fig. 175) represent any two complex numbers whose differ- 
ence OQ - OP is to be found geometrically. Since OP + OP' = OQ, 
OQ-OP = OP' = PQ. PQ, or its equal OP', represents then the 
required difference, in words, — 

The difference OQ — OP between two complex numbers, represented 
by OQ and OP respectively, is represented by the third side PQ of the 
triangle of which the other two sides are OP and OQ. 

148. Application of Complex Quantities to Physics. It is 

shown in physics that if OP and OP' (Fig. 175) represent in length 
two forces acting in the directions indicated by the arrows, then OQ, 
the diagonal of the parallelogram of which OP and OP' form the 
sides, will represent their resultant, both as regards magnitude and 



284 



PLANE TRIGONOMETRY 



LCHAP. XIV 



direction. It follows that if OQ represents a given force in magni- 
tude and direction, and OP its component in the direction OP, 
PQ = OP' must represent the other component. The laws of com- 
position and resolution of forces are then precisely those which govern 
the geometrical addition and subtraction of complex numbers; we 
may therefore compound or resolve forces by adding or subtracting 
the complex numbers which represent them. 

Suppose n forces /1, f 2 , / 3 , 
. . . fn (Fig. 176) act in the 
same plane on the same point 
P with the respective inten- 
sities ri, r 2 , r 3 , . . . r n pounds, 
and at angles 0i, 6 2 , 3 , . . . On 
with the x-axis. The forces 
are then represented by the 
complex numbers: - 




Fig. 176. 



/1 = y\ (cos 6 1 + i sin 0i), 
f 2 = r 2 (cos 6 2 + i sin 2 ), 
/ 3 = r 3 (cos d 3 + i sin 3 ), 



fn = r n (cos 6 n + i sin 6 n ). 
Adding all these quantities we obtain for the resultant: 

F = r 1 cos 0i + r 2 cos 2 + r 3 cos 3 + . . . + r n cos 

+ i (ri sin 0i + r 2 sin 2 + r 3 sin 3 + . . . + r n sin n ) 
= R (cos <f> + i sin <£), 

where i? is the modulus and cf> the argument of the complex num- 
bers representing the resultant force F. ■ 

The resultant may be found geometrically by constructing OA 
equal and parallel to r 1} AB -equal and parallel to r 2 , BC equal and 
parallel to r 3 , and so on. Then OD, the directed line joining the 
origin to the terminal extremity of the broken line thus con- 
structed, represents the resultant force F both as regards magni- 
tude and direction. 

We will apply the method just explained to the solution of Prob- 
lem 16, Exercise 26. Taking AP for the direction of the x-axis we 
have /"w , 



149] COMPLEX QUANTITIES 285 

ri=i5,r 2 = 6, ^3 =5-7, r ± = 7-9, ^=12.3, r 6 = 10. 

0i = o°, 6> 2 = 12 30', d 3 = 31° 21', 4 = 47° 46 r , ^5 = 58° 10', 6 = 72° l8'. 

/i = 15 (coso° + zsino°) = 15.0000 + i 0.0000 

/ 2 = 6 (cos 12 30' + i sin 12 30') — 5.8578 + 2 1.2984 

fz = 5.7 (COS31 21' + i sin3i°2i / ) = 4.8678 + 2 2.9657 

U = 7-9 ( cos 47° 46' + a sin 47 46') = 5.3104 + i 5.8492 

/ 5 = 12.3 (cos 58 io' + i sin 58 10') = 6.4882 + 210.4501 

/ 6 = 10 (cos 72 18' + 2 sin 72 18') = 3.0400 + 2* 9.5270 

Adding 

F = R (cos <^> + 2 sin 0) = 40.5642 + 2 30.0904 

£ = V40.5642 2 + 30.0904 2 = 50.51, 

, _ T 40.^6 . _ T 30.00 , / 

9 = cos * - — a - = sin * ^ — - = 36 34 , 

50.51 50.51 

whence F = 50.51 (cos 36 34' + i sin 36 34'), 

that is, the resultant force is 50.51 pounds and acts at an angle 
36°'34 r with AP. 

149. Historical Note. The method of representing complex 
numbers by points in a plane is often referred to as Argand's repre- 
sentation, after J. R. Argand, a French mathematician, who gave a 
discussion of the method in 1806. This is another misnomer, for it 
is now known that Caspar Wessel, a German, published the same 
method as early as 1799. The method was, however, completely 
forgotten until in 1831 it was rediscovered and applied by the great 
Gauss, to whom is generally conceded the credit of having established 
the true theory of imaginary numbers. To Gauss we owe the term 
"complex numbers" and the symbol i to represent the imaginary unit. 
The term " imaginary " was first used by Descartes. The choice of the 
term was unfortunate, for imaginary numbers as now understood 
are no more " imaginary " in the ordinary meaning of that term 
than are negative numbers. In view of their geometrical interpre- 
tation the name " lateral numbers " has been suggested in the place of 
the name " imaginary." The trigonometric form of complex num- 
bers was first used by the great French mathematician Cauchy. 
The concept of complex numbers and their geometrical representa- 
tion forms the basis of many of the branches o r higher mathematics, 
and is indispensable as well to the study of theoretical physics. 



286 PLANE TRIGONOMETRY [chap, xiv 

Exercise 62 

1. Represent geometrically each of the numbers, 

2, -3? 4 h - h 1 + i, 1 - i, 2 + 3 i, - i - i. 

2. Represent geometrically each of the numbers, 

2 (cos 30 + i sin 30 ), 
cos 6o° + 2sin 6o°, 3 (cos 120°+ i sin 120 ), \ (cos 240°+ isin 240 ), 

o 1 • • o o 1 • • o * /""* / 7T 1 • ■ 7T\ 

cos o + 1 sin o , cos 90 + 1 sm 90 , V 2 cos - + 1 sm - • 

V 4 4/ 

3. Express the following numbers in the trigonometric form: 
1V2+HV2, J+*jV3, l-*i^3, §^3+K 

^4?w. cos 45 + i sin 45 , cos 6o° + i sin 6o°, . . . , cos o° + * sin o°. 

4. Express each of the numbers in Problem 2 in the form x-\- iy. 

Ans. V3 + i t § + ij \/ 3 , . . . ; , 1 4 *. 

5. Compare the moduli of the following numbers: 

^3 + h ^3 — h —^3 + *> —^3 — h v / 2 + * ^2, ^2 — i V 2 , 

— V2 + «V2, — V2 — iV2, i + iV3, i — i\/^ } —i-\-i\Z^ } 

— 1 — i v 3, 2, 2 i, — 2, — 2 i. 

What conclusion can you draw with reference to the location of 
the points representing these numbers ? 

6. Add geometrically 1 + i and 2 + i; 

1 -f- 2 i, 2 + f, and 1 — i; 1 + 2 i, — 2 + 3 i, 1 — i, — 3 — i, and 3 — 3 i; 
1, i, 2, and 1 — i. 

7. Add 1 + 2 j, 2 + i, and 1 — i geometrically, in three different 
ways: 

(a) By adding the first and second and then the third, 

(b) By adding the first and third and then the second, 

(c) By adding the second and third and then the first. 

8. Subtract geometrically 2 + 3 i from 3 + 4 i; 2 — i from 1 + i; 
3 + 2 i from 1 — i. 

9. Prove the rule for geometric subtraction separately by means 
of the relation 

(x + iy) — (x f + iy') — (x — x') + i (y — y f ). 



ISO] • COMPLEX QUANTITIES 287 

10. Prove that the sum of the complex numbers representing the 
sides of a polygon taken in order equals zero. 

n. Three forces of 151, 106, and 61 horse power respectively, 
make angles of 50 04/ 30", 211 20' 305", and — 96 respectively 
with the x-axis Show that the resultant is zero, that is, that the 
forces are in equilibrium. 

150. Multiplication and Division of Complex Numbers. Let 

Z\ = r\ (cos 0i-\-i sin 0i), z 2 = H (cos 2 + i sin 2 ) 
represent any two complex numbers. Their product is 
^1^2 — ^2 (cos 0i + i sin 0i) (cos 2 + i sin 2 ) 

= rir 2 [ (cos 0i cos 2 — sin 0i sin 2 ) + i (sin 0i cos 2 + cos 0i sin 2 )] 
= r ± r 2 [ cos (6i + 6 2 ) + i sin (6i + 6 8 )]. (1) 

Dividing %\ by z 2 we obtain 

»j _ r\ (cos 0i + i sin gi) __ n t cos 0i + j sin 0i ( cos 2 — i sin 2 
« 2 H (cos 2 + i sin 2 ) r 2 cos 2 + i sin 2 cos 2 — i sin 2 
_ r\ (cos 0i cos 2 + sin 0i sin 2 ) j- z (sin 0i cos 2 — cos 0i sin 2 ) 
r 2 (cos 2 — i 2 sin 2 0) = 1 

= - 1 [cos (0! - 2 ) + i sin (Gx - e 8 )]. (2) 

From equations (1) and (2) it appears that, 

The product of two complex numbers is another complex number 
whose modulus is the product of the moduli and whose argument is the 
sum of the arguments of the numbers. 

The quotient of two complex numbers is another complex number 
whose modulus is the modulus of the dividend divided by the modulus of 
the divisor, and whose argument is the argument of the dividend dimin- 
ished by the argument of the divisor. 

Corollary 1 . Since 1 = cos o° + i sin o°, 

i = cos 90 + i sin 90 , 
— 1 = cos 180 + i sin 180 , 
the modulus in each case being unity, it follows that: 

multiplying any complex number by 1 leaves it unchanged, 

multiplying any complex number by i increases its argument by oo°, 

multiplying any complex number by — 1 increases its argument 
by 180 . 



288 PLANE TRIGONOMETRY »[chap. xiv 

In the first case the line segment representing the complex number 
is left unchanged, in the second case the line segment is turned in 
the positive direction through an angle of 90 , in the third case the 
line segment is reversed. 

Corollary 2. (cos + i sin 0) (cos — i sin 0) = 1; therefore 
cos + i sin and cos 6 — i sin are reciprocals, and in general the 

reciprocal of r (cos + i sin 0) is - (cos — i sin 0). 

r 

Two numbers such as r (cos + i sin 0) and r (cos — i sin 0) are 

said to be conjugate to each other. Each is called the conjugate of 

the other. 

151. Powers of Complex Numbers. By (1), Article 150, we have 

Z1Z2 = f\ (cos 0i + i sin 0i) X ^2 (cos 6 2 + 2' sin 2 ) 
= iv, [cos (0i +0 2 ) + * sin (0i+ 2 )]. 

Let us multiply this result by a third complex number 23, thus: 
Z1Z2Z3 = rir 2 [cos (0i + 2 ) + i sin (0i + 2 )] X n (cos 3 + i sin 3 ) 
= ?W3 [cos (0i + 2 + 3 ) + i sin (0 X + 2 + 3 )]. 

Similarly we obtain for the product of n factors 

Z\ = r\ (cos 0i + i sin 0i), z 2 = r 2 (cos 2 + i sin 2 ), . . . 

z n = ''n (cos 6 n + i sin W ) 

• ziz 2 . . . z n = rtfi . . . r n [cos (0i + 2 + . . . + d n ) 

+ isin (0i + 2 + . . . +0J] (1) 

From (1) we see that, 

The modulus of the product of any number of complex numbers is 
equal to the product of the moduli of the factors, and the argument of 
the product is equal to the sum of the arguments of the factors. 

Now let us suppose that the n factors in (1) are all equal, each 

factor being 

z = r (cos + i sin 0), 

we then have z n = r n (cos n + i sin n 0). (2) 

In particular, if r = 1, 

(cos 6 + i sin 8) rt = cos n 6 + i sin n 8. (3) 



r — i, cos0 = -, sin0 = — ^, hence = -> 



152] COMPLEX QUANTITIES 289 

Equation (3) embodies one of the most famous theorems of modern 
analysis. It is known as DeMoivre's* theorem, after its discoverer, 
and may be stated thus: 

The argument of the nth power of any complex number is equal to 
n times the argument of the number. The theorem may be shown to 
hold for any value of n, negative, fractional, irrational or even 
imaginary, but unless n is integral cos n + i sin n represents but 
one of the several values which (cos + i sin B) n may have. 

To illustrate the use of DeMoivre's theorem, we will employ it to 

raise - -\ — v 3 to the 9th power. Changing — | — V 3 to the 
22 22 

trigonometric form we have 

1 

2' 2 3 

and 

/i , t\/"V / 7T , • • 7r\ 9 Q1T , • • 97T 

( - + - V 3 = cos - + 1 sin - ) = cos 2 — h * sin ^— = — 1. 
\2 2 / \ 3 3/ 3 3 

Similarly 

(1 + i) 5 = V2 [cos-+ i sin- ) = 4V2 cos ^ + i sin ^ 
L V 4 4/J L 4 4 J 

= - 4 (1 + *)- 

152. Roots of Complex Numbers. Suppose it is required to 
find the nth. root of the complex number z = r (cos 6 + i sin 6). 

Denote the root \jz by 

2' = /(cos0' + isin0')- 

Then {z') n = 2, and we have by DeMoivre's theorem 

[/ (cos 0' + i sin 0')] n = r' n (cos n 6' + i sin nd') = r (cos + ** sin 0), 
from which 

/— %}, fid' = 0, 0+ 27T, + 47T, + 6T, . . . 0+ 2#7T, 

since when an angle is increased by any number of times 2 t both the 
sine and cosine remain unchanged. It follows that 

* DeMoivre (166 7-1 754) created a large part of that portion of trigonometry 
which deals with complex numbers. His death has a curious psychological in- 
terest. Shortly before his death he slept a little longer each day, until when the 
limit of twenty-four hours was reached, he died in his sleep. 



290 PLANE TRIGONOMETRY [chap xiv 

/ nr , + 27T d + 4ir + 6x e + k-n- 

n n n n n 



and 



/ n i~ ~l 1 0\ 

z = yz = r n l cos - + 1 sin - ) 
V » n/ 

= rn ( 



+ 2 X , . . + 2 X 

cos — ! h * sin 

w n 

-I + A X . .. + 4 x 

= r n ( cos — — h ^ sin — —^~ 

n n 



*-/ 0+2&X , . • 0+ 2k^ 

= m cos — ! h 1 sm 



n n I 

where k is any integer. 

These values are not all different, for when k = n we have 
cos + 2 rnr 



r 



,..0 + 2mr\ ^T fd , \, . . /0 , \1 
+ ^sin — ! \ = r n \ cos I — h 2x1+ 1 sinj - + 2x) 



n 

1 



, . . 0\ 
= r n cos — h « sin - > 

\ n nj 

and similarly 

If 6 + 2(n+l)w , . • 0+2p+l)x \ V + 27T , • . + 2x\ 

r n cos— J — - — ! — — +^sin — ! — - — ! — — )=rn cos— ! Hsin— ■ > 

\ n n J \ n n ) 

and generally 

-/ 0+2(w + ai)x 1 • • + 2(71 + ju)x\ 
r n cos — ! — - — L - — — h jsm — L — * — ^' 
\ n n } 



- f + 2 [X-K - . + 2 jU7T 

= r n cos — ! h sm — ! 



n n 

so that z f has only w distinct values corresponding to the values 

k = o, 1, 2, . . . n — 1, 

that is, 

Every complex number r (cos 6 -\- i sin 0) ^as w w2/f r00fo gww y 
//?e formula 

r / /1 1 • • /i\l~ ~ 0+2 &X ■ . . + 2 &xl / \ 

[r(cos0 + ism0)]^ —r n cos — ! h^sin — ! > (1) 

L n n J 

where k has the values o, 1, 2, . . . n — 1. 



152] COMPLEX QUANTITIES 291 

Example i To find the fourth roots of 1. 

Solution. In the trigonometric form 1 = cos o° + i sin o°, hence 
by the formula (1) above 

*/" /o + 2 kr 1 • • o + 2 kir \ 1 t 

Vi = cos [ — h t sm — ! , where k = o, 1. 2, or 3. 

V 4 4 / 

Denoting the four roots by z , Zi, Z2, z 3 respectively, we have 

0...0 27T,..27r. 

z = cos - + 1 sm - = i, Z\ = cos 1- t sm — = i y 

4 4 4 4 

ait , • . Air 6ir , . . 6ir 
z 2 = cos - — h^sin— =-1, z 3 = cos \- 1 sm — = — 1* 

4 4 4 4 

Example 2. To find the fifth roots of — 1. 

Solution. In the trigonometric form — 1 = cos w + i sin w, hence. 

5 1 7T + 2 klT - ..7T+2&7T -i 7 

V— 1 = cos h ^ sm — ! , where & = o, i, 2, 3, 4. 

5 5 

This gives the five values 

z = cos- + 1 sm-, Zi = cos 52 \- 1 sm" 2 — , z 2 = cos 71—}- ^sinx, 

5 5 5 5 

77T ■ • • 77T Q7T , • • Q 7T 

Z3 = cos ■* r 1 sin ■*— , z 4 = cos 2 1- 2 sm *- • 

5 5 5 5 

The last two roots may be written equally well 



Z3 = cos * 1 sin i2 — , z 4 = cos ^ sin - » 



■*— =cos( 2 7T— j2 — ^cos 22 — , sinl_ =sm(27r — j2 — )=— sin* 2 — j 
5 "V 5/ 5 5 V 5/ 5 



for 

cos J -- 

and 



(7r\ 7T . Q7T . / 7r\ . 7T 

2 7r = cos - , sin 2— = sm [ 2 r ) = — sin - . 

5/5 5 \ 5/ 5 

so that finally 



cos^— = 
5 



5/ 7T , • • 7T 3 7T , • • 3 7T i • • 

V — 1 = cos - ± * sin - , cos s2 — ± t sm j2 — , cos ir + * sin 7r= — 1. 

5 5 5 5 



292 PLANE TRIGONOMETRY [chap, xiv 

153. To Solve the Equation z tl - i = o. If z n - i = o, then 

z n = i = cos o + i sin o, 
and the w roots are 

o , . . o 
z = cos — h i sin - = i, 

n n 

Z\ = cos h ^ sin — , 

n w 

at ■ . . AT 

z 2 = cos (- i sin s— > 

w w 



Now 



^ 2(«-2)ir | • • 2 (w — 2) T 

z n -2 = cos — — -j- % sin — '— j 

n w 

2(n—i)ir - • . 2 (w — iW 
z n _! = cos — - h 1 sin — * '- — 

w w 

2 (fl — l)lT I 2t\ 2 7T 

COS — - — = COS ( 2 7T = COS — t 

n \ n ) w 

. 2 (w — iW • / 2 7r\ • 2 7T 

sin — - — = sin 2 t = — sin — > 

w \ n I w 



hence 



27T ..27T J* Ml 47T ..47T 

z n -i= cos 2 sin — , and similarly z n - 2 = cos- % sm=^-> 

ww ww 

that is, Z\ and z n - 1 are conjugate complex numbers, and likewise 22 
and z n -2, Zz and z n _ 3 , etc., are each pairs of conjugate numbers. 

(a) Let w = 2 m + 1, an odd number. Then besides the first 
root z there are w pairs of conjugate roots, that is, the roots are 

27r ( ..27T AT . * > AT 2 7W7T ... 2 WMT 

i, cos — ± i sin — , cos — ±?sin— , . . . cos ± i sin 

w w w w w w 

(b) Let « = 2 w, an even number. Then 

2 7W7T ... 2 7W7T , • • 

z n = cos h ^ sin = cos7r + i sin7r = — I, 

w w 

and the roots are 



27T , • • 27T 47T , . . ATT 

i, cos — ± i sin — , cos tt — ± ^ sin 31 — , 
w ww w 



4jr 

W 
(m— l)2 T , • • (W — l)2 7T 

* - ± i sin- — , — 1. 



cos 

w w 



154] COMPLEX QUANTITIES 293 

The roots of the equation z n — 1 = o are represented geometri- 
cally by the n lines (or by their terminal points) drawn from the 
origin as center to the circumference of a circle, radius unity, so as 
to divide the circumference into n equal parts, one of these lines 
coinciding with the positive direction of the x-axis. 

154. To Solve the Equation z n + 1 = o. If z n + 1 = o, then 

z n = — 1 = cost + i sinr, 
hence the n roots are 

7T - • • 7T 

20 = cos — h 1 sin - , 
n n 

Z\ — cos s2 h 1 sin Q — ? 

n n 

Kir , • • <tt 

Zi = cos J h ^ sin »*— , 

w w 

z n _ 2 = cos * ^ h z sin i s2Z - , 

n n 

(2W— iW , . . (2H— i)ir 

z n - 1 = cos ~ r * sin * ^— , 

w w 

where 20 and z n -i, Z\ and z n _2, etc., are pairs of conjugate roots. 
(a) Let n = 2 m + 1, an odd number. The middle root is 

2 m = cos * ! — L h * sin a ! — '— = cost + 1 smir = — 1, 

w w 

so that the n roots are 

cos - ±?sm-, cos Q — ± z sm s2 — , . . . , 
www w 



(2 m— iW . . . (2 m— iW 

* — ±*sm- — , —1. 



cos 

n n 

(b) Let n = 2 m, an even number. Then the roots are 

7T...7T 3 7T . • • 3 7T 

cos - ± 1 sm - > cos Q — ± ^ sin i2 — > . . . , 

n n n n 

(2m—i)ir , • • (2m— i)ir 

cos — ± ^ sin - — 

w n 



294 PLANE TRIGONOMETRY [chap, xiv 

The roots of the equation z n + i = o are represented geometri- 
cally by the n lines (or by their terminal points) drawn from the 
origin as center to the circumference of a circle, radius unity, so as 
to divide the circumference into n equal parts, the positive x-axis 
being taken to bisect the angle between a pair of consecutive lines. 

Exercise 63 

Compute the following expressions by DeMoivre's theorem, then 
verify your results by expanding the binomials by the binomial 
theorem. 

1. (1 + i) 2 . Ans. 2 i. 

2. (1 — iy. Ans. — 4. 



3- 


(;+^)" 


4- 


A/ 2 iV 2 \ 2 




fV-i i \ 5 



Am. — i. 
\ 2 2 / 

6. Show that each of the following expressions equals — 1, 

( cos - ± 1 sin - > cos ^— ± % sin v — > (cos x ± 1 sin irr. 
\ 5 5/ V 5 5 / 

7. Construct the points representing the expressions 

(cos- + i sin -)> (cos- — i sin-)> etc., in Problem 6. 
\ 5 5/ V 5 5/ 

8. Find the seven seventh roots of 1 and plot the points represent- 
ing these roots. 

A 27T , • • 2 7T ATT . • • ATT 6 7T , • . 6 7T 

Ans. i, cos — ±i sin — , cos 3 — dz i sin 31 -, cos — - ± i sin 

7 7 7 7 7 7 

9. Find the seven seventh roots of — i and plot the points represent- 
ing them. 

10. If z , z-ij Z2, . . , z TO -i are the roots of the equation z n — i = o, 
show that 

Z 2 = Zi 2 , Z 3 = Zl 3 , 24 = Zl 4 , . . . Z n -i = Zi n_1 . 

11. Find all the values of Vi + i and represent them geometri- 
-cally. Ans. V2 (cos 9 + i sin 9 ), V2 (cos 8i° + i sin 8i°), etc. 



I56J 



COMPLEX QUANTITIES 



295 



155. The Cube Roots of Unity. Let u , u h u% represent the three 
cube roots of 1, then by the preceding article 

Uo = cos o + i sin o = 1, 



27T 1 • • 2 7T — I 

U\ = cos h ^ sin — = 

3 3 2 

4T 1 • . Alt — I 

2*2 = cos - — h * sin = L - = 

3 3 2 



+ * 



2 

V- 



Now 



2 7T Y _ 4 7T 



W1 2 = ( cos h i sin — ) = cos — + * sin fLiL = «2, 

V 3 3 / 3 3 

o / 4 7T - • • 47r\ 2 2 7T - • • 2 7T 

W2 2 = cos - — h % sin — = cos \- 1 sin — = uu 

V 3 3 / 3 3 

also W1W2 = W1 3 = u£ = 1, 

that is, 

TTfe square of either of the imaginary cube roots of unity equals the 
other ■, cmJ //zew' product equals unity. 

We may then denote the cube roots of unity by 

I, CO, CO 2 , 

where co is either one of the roots Uu u±. 



156. The Cube Roots of Any Real or Complex Number. 

Let r (cos + i sin 0) represent any number and z Q , Zi, z 2 its cube 
roots. 
We have 



z = r* ( cos - + i sin - ]> 
3 3 



Sl 



= ,*/ 



+ 27T , • • + 2 

cos — ! h fc sin 



! ) 



= r s ( cos — h ^ sm - I cos \- t sm — = coz , 

V 3 3/V 3 3/ 

(• 



J + ATT , • • + 47T 

2 2 = r* ( cos — L - ii h a sm 



= r 3 f cos - -f- t sm - j ( cos \- 1 si 

where co has the meaning given it in Article 155. 



n — )= C0 2 2o, 



Y (1) 



296 PLANE TRIGONOMETRY [chap, xiv 

Furthermore, by applying the results just obtained 

COZi = C0 2 2 = 2 2 , C0 2 2i = CO 3 2 = ^0, (2) 

C02 2 = CClho = 2o, C0 2 2 2 = C0 4 2 = ^0 == 2l- (3) 

From (1), (2) and (3) it appears that any two cube roots of a 
number may [be obtained by multiplying the third by co and co 2 
respectively. Thus: 

from (1), from (2) from (3), 



2i = C02o, 22 = 0)2i, 2 = C022, 
2 2 = C0 2 2 , 

where 00 = — - ± i — 2 



2 2 = C0 2 2 , 2 = C0 2 2i, Z\ = C0 2 2 2 , 



157. Solution of Cubic Equations. Every cubic equation can 
be expressed in the form 

<*>oO? + 3 a i™ 2 + 3 a 2™ + «3 = o, (1) 

where a , a,\, (h, #3 are given numbers. 

Equation (1) can be replaced by another in which the second 
term is missing by putting 

x = -• (2) 

a 

On making this substitution, equation (1) reduces to 

2 3 + 3 Hz + G = O, (3) 

where 

fl" = a O2 — ai 2 , G = a 2 a 3 — 3 a aiO2 + 2 fli 3 . (4) 

Now put 

z = u + v, (5) 

then (3) becomes 

(u + z;) 3 + 3 #0 + »)+ G = ^ 3 + *> 3 +30 + v)(#+««0+ G=o. (6) 

If furthermore we put 

# + uv = o, that is, uv = — H, (7) 

then (6) becomes 

w 3 + a 3 = - G, (8) 

and substituting for v in (8) its value from (7) 

H 3 

u z = — G, or u 6 + G« 3 — H z = o. (0) 

w 3 



i 5 7] COMPLEX QUANTITIES 297 

(9) is a quadratic equation in u 3 . Solving 



3 - G± VG 2 + 4Hz 

u 6 = - — 1 

2 



tf = — G — u z 



so that for either sign we obtain 



-G=F VG 2 + 4g» 



==u + v== J-g + Vg 2 + 4 h^J-g-Vg 2 + 4 h^ 
* 2 * 2 

o- rr 

From (7) i> = , so that (10) may be written z = u , 

u u 



do) 



where u = y — — 



But the cube root of every number has three values, which by Article 
156 may be denoted by u, cou, cc 2 u respectively, where u is any one 
of these roots. The three values of z which satisfy the equation (3) 
are therefore 

z = u 9 

u 

H <* 2 H , v 

z x =<&u — — = <au 9 (11) 

(OU u 

o H 2 <*H 

z % =<aru — - = art* 9 

(0% u 



u being either one of the cube roots of — ^— = (12) 

2 

In applying this method to the solution of any equation of the 
form (1), we first find H and G from (4), then u from (12), then the 
three values of z from (11) and finally the three corresponding 
values of x from (2). 

Example i. Solve the equation 8 x* + 12 x 2 — 42 x — 95 = 0. 
Solution. Here ao = 8, #i= 4, (h = — 14, a 3 = — 95. 

From (4), 

H = aod2 — ai 2 = — 128 = — 2 7 , 

G = ao 2 a s — 3 0o0i02 + 2 01 3 = — 4608 = — 2 9 3 2 . 



298 PLANE TRIGONOMETRY [chap, xiv 

From (12), 

VG 2 + 4# 3 = V2 18 3 4 - 4.2 21 = 7.2 9 , 



w 



= ^/ -G-j-VG 2 + 4 H 3 = / Ms 2 + 7-2 9 = 24> 



From (11), 

Zo= U = 2 4 + 2 3 = 24, 

U 



Zi= cow ■ = 16 ( h 1 — ^ 1+8 f 1 — a )= — 12+4 *v 3, 

U \2 2 / \ 2 2/ 

W \ 2 2/ \2 2/ 



From (2) 

gb-fli _ 24-4 _ 
*°~ a " 8 _2 ' 5 ' 

* 1= »i-*i = -i2+4^V3-4 = _ 2 + ^ 
6Z 8 2 

% = »-* = -12-4^3-4 _ _ 2 _ y3 i: 

a 8 2 

Check. ( x + 2 2 ^ j J x _j_ 2 -| 2. i ) = # 2 -}- 4 # -f- - 1 /- = o, 

(x 2 + 4* + -\ 9 -) - f) = r* + f x 2 - -V-x - V- = o, 
or 8 # 3 + 12 x 2 — 42 x — 95 = o. 

158. The Irreducible Case. When G 2 +4 H 3 is positive, as in the 
preceding example, its square root is real, and u, which is the cube 

root of — ^—^ — can be found by the rules of arithmetic. 

2 

But if G 2 + 4 H 3 is negative, its square root will be imaginary, and 
we must employ DeMoivre's theorem to find u. This is the so-called 
" irreducible case " of the cubic equation. Its solution is as follows: 
Since G 2 + 4 H 3 is negative, — (G 2 + 4 H 3 ) will be positive, and 
we may put 

, -G + iV- (G 2 + 4H 3 ) , a , . . Q s 
u 3 = ! — t — - = r (cos 6 + i sin 6) , 



158] COMPLEX QUANTITIES 299 

whence, by Article 146, (3), 

r = J{-Gy + \-((?+im = Vlff, (l) 

* 4 

cos0 = ~ • (2) 

2V-H* 



u = r* 



u 



Hence, 



3 cos -+fcsin-J = V — £n cos - + * sin - ), 

— = V — H [ cos t sin - b 

' V 3 3/ 

»/ 77/ 0+27T , • • + 27r\ 

= V — // cos — ! (- ^ sin — ! , 

V 3 3 / 

-# ./ 7r( 0+27T . . + 27r\ 

= V — H ( cos — ! 1 sin — ! ], 

ww \ 3 3 / 

oj 2 w = V — /7 cos — L - a h * sin — - , 

V 3 3 / 

-£f ./ — nt + 47T . . + 4tt\ 
= V — H[ cos — —^ 1 sni — L - *— • 

co 2 w V 3 3 / 



^ */ — ^ 

g=M -- = 2V- ffCOS-, 

u 3 

JET . / — + 2 it , x 

«! = mi - — = 2 V - IT cos — 1 , (3) 

1 am 3 

H . / — + 4 TT 



and from Article 157, (2), 

z — a ± 



00 = 



a 



(4) 



Example i. Solve the equation 3 x 3 + 3 # 2 — 3 # — 2 = 0. 
Solution. a = 3, ai= 1, a 2 = — 1, a 3 = — 2. 

Z7 = a 2 ^2 — ^i 2 = — 4, G = a 2 a s — 3 a ai(h + 2 a/ = — 7, 
G 2 + 4 J3" 3 = — 207, hence we are dealing with an irreducible case. 
From (2), 

cos0 = ~ = '2 = °-4375, 9 = 64° 3' 20". 

2 v — H z 16 



300 PLANE TRIGONOMETRY [chap, xiv 

From (3), 



Z = 2 V — H COS - = 4 COS 21° 2l' 7" = 3.7256, 

3 

Z\ = 2 V— g cos 2 - = 4 cos i4i°2i / 7"=— 3.1241,, 

3 

z = 2\/^0 : cos^- : t4^=4COS26i o 2i , 7 ,, = -o.6oi5. 
3 
From (4), 

Zn — Q>\ Z\— a\ Z<l — CL\ 

X = — - 1 »i = — ■* #2 = — 

Gq CLq CLq 

= 0.9085, = - 1.3747, = ~ o-5338. 

Exercise 64 
T ,. - ~ * + *^3 M -- -!-*' ^3 

I. COi — , CO2 — . 

2 2 

By actual multiplication, show that 

COi 2 = CO2, 0>2 2 = «l, W1CO2 = I, CO] 3 =1, OJ2 3 = I. 

Show also that 1 + coi + W2 = o. 

2. Compute all the cube roots of 27; of 27 i. 

Ans. 3, - , 3 + 3<V3 , -3-3*^3 . 

2 2 

3^3+ 3 * y -3^3 + 3* ; _ 3 ^ 
2 2 

3. Compute all the cube roots of 1 + i: 

Ans. V2 [cos h * sin — ), V2 (cos 3^ + i sin — 

V 12 12/ \ 4 4/ 

6 f I 1 1 7T , . . 1 1 7r\ 

V 2 cos 1- t sin - 

\ 12 12 / 

Solve the following equations: 

4. a^— 6#— 9 = 0. Ans. 3, — *-= 3 # 

2 

5. 4 z 3 — 24 x 2 + 45 x — 25 = o. Ans. 1, 2.5, 2.5. 

6. :*? — 12 x — 10 = o. Ans. 3.8232, — 2.9304, — 0.8928. 

7. x* + 9 x 2 + 24 x + 19 = o. 

yl?w. By use of 4-place tables, —1.4680, —4.8794, —2.6527. 



iS9] COMPLEX QUANTITIES 301 

8. A man invests $5000 and two years later $2000 more. The 
interest was added to the principal at the end of each year. At the 
end of the third year the total increase was found to be $2058.16. 
Find the rate per cent of profit. Arts. 11 %. 

(Suggestion. Let x = 1 + rate.) 

9. The inside of a tank is 3 ft. wider than it is deep and 3 ft. longer 
than it is wide. Its volume is 100 cu. ft. Find the dimensions of 
the tank. Ans. 2.284, 5-284, 8.284. 

10. A loan of $500 is to be repaid in three equal annual payments 
of $190 each without further interest. Find the rate per cent. 

11. In determining the deflection of a beam, uniformly loaded 
and supported at its two ends and points of trisection, the follow- 
ing equation occurs: 

20 x* — 24 x 2 + 3 = 0. 

Find its roots. Ans. 0.4460, 1.0687, ~" °-3 I 47- 

159. To Express sin n8 and cos nft in Terms of sin 8 and cos 8. 

DeMoivre's theorem enables us to express the sine and cosine of 
any multiple angle nB in terms of powers of the sine and cosine 
of 6. We need only compare separately the real parts and the 
imaginary parts of 

cos n$ '+ i sin n$ = (cos + i sin 6) n (1) 

after expanding the right-hand member by the binomial theorem. 
For shortness sake let us put 

cos 6 — c, sin 6 = s, 
the right-hand side of (1) then becomes 

( c +i s )n = c n+ nk n-l s + g fe~ i) fV _j ^2 _[_ g (g ~ i) (» ~ *) ^n-3^3 

I • 2 I • 2 «3 

+ n ( n - l} ( W - 2 ) ( W - 3) . 4cn _ 4si + 

1- 2-3-4 

n 1 ' n-1 n(n—l) n _ 2 2 tl (tl — l) (ft — 2) . n o, 

I • 2 1-2-3 

1 . 2 • 3 -4 
since f 2 = — i, & = — *, ^ 4 = i, etc. 



302 PLANE TRIGONOMETRY [chap, xiv 

The real part of this expression must equal cos nd, since this is the 
real part of the equivalent left-hand member in (i), and for a like 
reason the imaginary part must equal sin nB, hence we have 

cosnO^-^-'V-y 

I -2 

+ n(» - I) (n - 2)(n - 3) c ~-4 ^ _ etc . (a) 
I-2-3-4 

sin n0 = wc"- 1 * - "(»-')(»-») c ~-3 s s 

I «2»3 

+ n(n-i)(n-2)(n- 3 )(n-4) c »-5 s 5 _ etc# ( j 
1 -2-3-4-s 

Thus, if w = 2, 



= r2_ 2 (2 - 1) 
1 • 2 
sin 2 = 2 cs = 2 sin cos 0. 



cos 2 = c 2 ' c° s 2 = c 2 — s 2 = cos 2 — sin 2 0, 

1 • 2 



If rc = 3, 
cos 3 = c 3 — 2-^3 Z C5 2 _ ^3 _ 2 £$2 _ cos 3 _ ^ cos sin 2 

I -2 

= 4 cos 3 — 3 cos 0, 
sin 3 = 3 c 2 s — 2_i2 LLA3 / c o ^3 _ ^ C os 2 sin — sin 3 

1-2-3 

= 3 sin — 4 sin 3 0, 

results which agree with those obtained in Exercise 50, Problems 10 
and 11. 

160. To Express cos and sin 9 in Terms of Sines and Cosines 
of Multiple Angles. By Article 150, cor. 2, 

cos + i sin and cos — i sin are reciprocals. 

Put z = cos + i sin 0, then — = cos — i sin 0, 

z 

z n = cos nB -\- i sin w0, — = cos nB — i sin w0, 

z n 

hence 

2 H = 2 cos 0, z = 2 i sin 0, 

z z 

z n -\ = 2 cos nB, z n = 2 i sin w0. 

z n z w 



160] COMPLEX QUANTITIES 303 

Now let us expand lz-\ — J and (z — ) by the binomial theorem, 

(z+— V = 2 n COS n d = Z n +nZ n ~ 2 -f ^(^- I ) z n-4_|_ . . . 
\ Z ) I • 2 

,n{n— 1) 1 1 1.1 ( x 

H — ■ • — 7 + w > (i) 



I • 2 



,n — 4 «n-2 



/ 2 - - V = 2 n f" Sin n = Z n - W2 n ~ 2 + ^ ?) z n_4 _ . . . 

\ Z / I • 2 

. n(n — 1) 1^ 1.1 /x 

where in (2) the upper signs in the last terms are to be used when n 
is even and the lower signs when n is odd. 

Let us group together the first and last term in each of the ex- 
pressions (1) and (2), also the second and second last, the third and 
third last, etc.; we may then write 

+ nJ ~(^ + ^+--- (3) 
2 n i n sin n = (z n ± J ^)~ nlz n ~ 2 ± -± 

where in (4) the upper signs are to be used when n is even and the 
lower signs when n is odd. The total number of terms in each 
binomial expression is one more than the index n, and since we have 
grouped the terms in pairs, there will be one term left over in case n 
is even. This term will not contain z at all, for since the exponent of 
z diminishes by 2 for each successive term, it will be o, and z° = 1. 

Let us now substitute for z n -\ , z n , etc., their values 

z n z n 

2 cos n$, 2 i sin nS, etc., and divide out the common factor 2. This 
gives 

2 W_1 cos" 6 = cos nQ + n cos (n — 2) 6 

. n (n — 1) , N « . , / N 

_| 1 > C os {n - 4)6 + etc., (5) 

1 • 2 



304 PLANE TRIGONOMETRY [chap, xiv 

2 n ~ 1 i w sin w 9 = cos rS — n cos (n — 2)9 

+ IL&L^JLl cos (n- 4)0 -etc., (w even) (6) 

1 • 2 

= £ [sin w9 — n sin (n- — 2)9 

+ n ( n ~ J ) S in ( n - 4)6 ~ etc.,] (n odd) . (6') 

1 • 2 

The factor i disappears in either case, for when n is even, say 
2 m, we have $ n = i 2m = (— i) m , which is +1 or — 1 according as 
m is even or odd, and when n is odd, say 2 m + 1, we can divide 
both sides of the equation (6') by i and have left on the left side 
i 2 m = + 1 or — 1 as before. 

Examples. If n = 2, 

2 cos 2 = cos 2 0~+ 1, or cos 2 = — ^ 

2 

2 i 2 sin 2 = cos 2 — 1, or sin 2 = 

2 

If « = 3, 

«, o fl a, a <? a 3 COS + COS 3 

2 2 COS 3 = cos 3 + 3 cos 0, cos 3 = Q ! £- • 

4 

•>•*•■?/> •/• a -m -i/i 3 sin — sin 3 
2 2 r sin 3 = 1 (sin 3 — 3 sin 0), sin 3 = « *2 — 

4 

If « = 4, 

,4a />■ Ai/c'4/i 6 + 4 cos 2 +COS4 
2 3 cos 4 = cos 4 + 4 cos 2 + 6, cos 4 = — — ! — • 

■,-a • A n a q i z- • 4 /j 6 — A COS 2 + COS 4 

2V sm 4 = cos 4 — 4 cos 2 + 6, sin 4 = ! — • 

8 

Exercise 65 

1. By the method of Article 159 express cos 4 and sin 4 each in 
powers of sin and cos 0. 

^4w5. cos 40 = cos 4 0—6 cos 2 sin 2 + sin 4 0, 
sin 4 = 4 cos 3 sin — 4 cos sin 3 0. 

2. Show that 

a 4tan0 (1 — tan 2 0) 

tan 4 = — — • 

1 - 6tan 2 + tan 4 



i6o] COMPLEX QUANTITIES 305 

3. Show that 

cos 5 6 = cos 5 — 10 cos 3 sin 2 + 5 cos sin 4 0. 

= cos (16 cos 4 — 20 cos 2 + 5). 
sin 5 = sin 5 — 10 sin 3 cos 2 + 5 sin cos 4 0. 

= sin (9 (16 sin 4 — 20 sin 2 + 5). 

a tan (tan 4 - 10 tan 2 0+0 
tan c = * — -^ • 

5tan 4 0- iotan 2 + 1 

4. By the method of Article 160 express cos and sin in terms of 
functions of multiple angles. 

CO s 5 = 2OCQS + 5 cos 3 + cos 50 

16 

sin 5 = 2osin0- 5 sin 3 + sin «; 
16 

5. Show that 

cos 6 a _ 20 + 15 cos 2 + 6 cos 40 + cos 6 

32 

„• 6 a 20 — I 5 COS 2 + 6 cos 4 — COS 6 
sin = - ■ - • 

32 



CHAPTER XV 

TRIGONOMETRIC SERIES AND THE CONSTRUCTION OF TABLES 

161. Definition of Infinite Series. An infinite series is an indi- 
cated sum of an endless number of terms. Infinite series are of 
common occurrence in arithmetic and algebra. Thus in 

1 1,1,1, i , / T N 

- = o.iiiii . . . = 1 1 1 h . . . , (i) 

9 io ioo iooo ioooo 

the right-hand member is an infinite series. Similarly, every recurring 
decimal may be expressed as an infinite series. Again, if we divide i 
by i — x, according to the rule for long division we obtain 

= I + X + X 2 + X s + x 4 + . . . (2) 



1 — X 



The right-hand member of this is an infinite series. The square root 
of every number which is not a perfect square and the cube root of 
every number which is not a perfect cube may be expressed in the 
form of an infinite series. 

Since it is impossible to write down all the terms of an infinite 
series, the series is not completely determined until we know the law 
or rule according to which the various terms are formed. When 
this law is known we can write down any term that is needed. In 

the first series above, the law for the general term is — > and the 

5 io n 

series is completely expressed thus : 

i + -i- + -j-+ . . . +^-+ . . . 

IO IO 2 IO 3 io n 

In like manner the series (2) is completely expressed by 

I + X + X 2 + . . . + x n ~ l + . . . 

In each case n stands for the number of the term. 

When the law of formation is clearly apparent from the first few 
terms, as in the above examples, the general term is not always 

expressed. 

306 



i6 2 ] TRIGONOMETRIC SERIES 307 

A series is frequently expressed by writing the Greek letter 2 
before the general term; thus the second series above may be written 
2 x n ~ l , which is read " summation of x n_1 ." 

The general expression for an infinite series is 

2 U n = Ui + 1H + u z + • • • + u n + 
where u\, u%, Uz, . . . u n , are the terms of the series. 

162. Convergent and Non-Convergent Series. Let S n stand for 

the first n terms of the series, thus 

"5*2 = ^1 ~f* ^2j 

S3 = U\ + U2 + u 3) 

S± = U\ + ih + Uz + u^ 

S n = Ui + U2 + u 3 + • • • + w n . 

As w increases indefinitely (approaches <*), one of three things must 
happen, — 

(a) S n may approach some finite quantity as its limit. 

(b) S n may become larger than every assignable finite quantity. 

(c) S n may neither approach a finite limit nor become infinite, 
but fluctuate between two or more different values. 

No other alternative is conceivable. In the first case the series is 
said to be convergent, in the second divergent, in the third oscillating. 
Divergent and oscillating series are grouped together under the 
term non-convergent series. 

If a series contains a variable, as the series (2), Article 161, the_ 
series may be convergent for certain values of the variable and non- 
convergent for other values. 

Example i. Consider the series 

x + x 2 + :r 3 + ••• + % n + ••• 
(a) If %_=■ J, the series becomes 



2 2 



c Io I I I x C I L I i I x 

2 24 4 248 8 



c 1.1,1,1 I 1 1 1 

2 4 8 16 2 n 



n 



308 PLANE TRIGONOMETRY [chap, xv 

As n approaches oo , — approaches o and S n approaches i, hence the 

2 n 

series comes under (a) and is convergent. 

(b) Next let us put x = 2, then the series becomes 

2 + 2 2 + 2 3 + ••• + 2 n + ..." 

5i = 2, £2=2+4 = 6, S 3 = 2 + 4 + 8 = 14, 

S n = 2 + 4 + 8 + ; • • + 2 n . 

Plainly, as n approaches 00 , S n approaches 00 , hence the series comes 
under (b) and is divergent. ' 

(c) Finally, if x = — 1, the series becomes 

-1 + 1-1+ ... +(-i) n + ••. 

S\ = — 1, 6*2 = o, S3 = — 1, S4 = o, S n = — 1 or o according 
as n is odd or even. In this case S n neither approaches a finite 
limit nor becomes infinite, hence the series is oscillating. 

Divergent series frequently lead to absurdities, and must therefore 
be avoided. For instance, if in the series (2), Article 161, we put 

x = 2, the left member becomes = — 1, while the right member 

1 — 2 

becomes 

1 + 2+4 + 8+ ... ; 

hence for the value x = 2, which makes the series divergent, the 
two members of (2) can no longer be considered equal. 

163. Absolutely Convergent Series. A series which remains 
convergent after all its terms are made positive is said to be abso- 
lutely convergent. 

Convergent series which become divergent when all the terms 
are made positive are said to be semi-convergent or conditionally 
convergent. 

For instance, if in (2), Article 161, we put x — — j, the series 
becomes 

1 1 1 x _i_ 

1 h — :+ • • •, 

2 2 l 2 6 

which is convergent, for it remains convergent when all its terms 
are taken with the positive sign. The series is therefore absolutely 
convergent. 



163]' TRIGONOMETRIC SERIES 309 

The series 1 — | + 3 — i+ • • • is convergent, but when all 
the terms are made positive the resulting series is divergent. The 
given series is therefore a semi-convergent or conditionally con- 
vergent series. 

Absolutely convergent series are subject to all the fundamental 
laws of algebra,* that is, they may be added, subtracted, multiplied 
and divided, like expressions consisting of a finite number of terms. 
This is not true of semi-convergent and divergent series. Curious 
results may be arrived at if this is not kept in mind. For example, 
take the divergent series 

■S = 1 + 3 + 5 + 7+9 + 11 + 13 + 15+ ••• 
0=1—1 + 1— 1+ 1 — 1+ 1— 1 + • • • 

Adding 5=2 + 2 + 6+ 6 + 10 + 10 + 14 + 14 + • • • 
.= 4 +12 +20 + 28 + • • • 

= 4(1 + 3 +5 + 7 + • • • ) 

= 4 S, which of course is absurd. 

Or take the semi-convergent series 

kS=I _I + I_I + I_I + ... 
2 3 4 5 6 

which is known to be equal to log 2 = 0.69315 • • • We may write 

5 =I +I-l + I + I-£ + I + I-f+ ... 
22344566 



,1,1,1,1, (2 

2 3 4 5 \2 



+-+f +^+ 

4. 6 8 



= I +I+I+I+*+ .-. -( I+ I + I + I + ..- 
2345 \ 2 3 4 

= o, that is, a constant 0.69312 • • • equal to zero, which 

is absurd. These examples show that we cannot treat an infinite 
series like we do other expressions until we know whether it is abso- 
lutely convergent or not. 

* For the proof of this statement we must refer the student to textbooks on 
higher algebra, such as Chrystal's Algebra, Chapter XXVI. 



3IO PLANE TRIGONOMETRY [chap, xv 

A series which is absolutely convergent will of course remain con- 
vergent when some or even all of the terms are made negative, but 
if a series is divergent when all its terms are positive it may become 
convergent when a certain proportion of its terms are made negative. 

164. The Sum of an Infinite Series. When a series converges, 
not otherwise, the limit which S approaches as n approaches oo is 
called the sum of the series. Thus when we say that the sum of the 
series 



I + I + I+:. 

248 


• + -+.•• 

2 n 


is 1, we mean that the limit of 




S.-I + I + I + . 

248 


2 n 2 n 



is 1 as n approaches 00 . A divergent series has no sum in the proper 
sense of that word. 

165. The Limit of r n as n Approaches 00 . 

(a) Let r = 1. 1 multiplied by 1 equals 1 no matter how often 
the multiplication is repeated. Hence the limit of r n = 1. 

(b) Let r > 1. We may write r = 1 -\- d, where d is some posi- 
tive quantity. By applying the binomial theorem we have 

r n = (1 + d) n = 1 + nd + ■ • •, 

Now no matter how small d may be, if n is taken sufficiently large nd 
can be made larger than any assignable quantity, hence we see that 
as n approaches 00 , r n approaches 00 also. 

(c) Let r < 1. Then - > 1, and since by (a) and (b) (- f = - 

r \r ) r n 

approaches 00 as n approaches 00 , therefore r n approaches — = as 
n approaches 00 . 

As n approaches 00 , r n approaches o, 1 or 00 according as r is less 
than, equal to or greater than 1. 

166. The Infinite Geometrical Series. Let 

S n = a + ar + ar 2 + . • . + ar n ~\ 
then rS n = ar + ar 2 + . . • + ar n ~ l + ar n . 



167] ' TRIGONOMETRIC SERIES 311 

Subtracting S n — rS n = (1 — r) S n = a — ar n — a (1 — r n ), 

from which S n = — — • 

1 — r 

(a) Let r be numerically less than 1. By Article 165 (c) as n 

approaches 00 r n approaches o, hence S n approaches the limit 

1 — r 

and the series is convergent. Since the series converges when all its 
terms are positive, it is absolutely convergent. 

(b) Let r = 1. Then S n = a + a + & + • • • +# = na, which 
approaches 00 as n approaches 00 ; hence in this case the series is 
divergent. 

(c) Let r = — 1. Then S n = a — a + a — • • • +(— i) n_1 a = o 
or a, according as n is even or odd. In this case the series is 
oscillating. 

(d) Let r be numerically greater than 1. By Article 165 (b) as n 

approaches 00 r n approaches 00 , hence S n — — approaches 00 

1 — r 

and the series is 'divergent. 

The results may be summed up in the following theorem: 

An infinite geometrical series is absolutely convergent if its ratio r is 
numerically less than 1, non-convergent if its ratio is numerically equal 
to or greater than 1. When convergent its sum 

e _ a 
1 — r 

where a is the first term and r the ratio. 
167. Convergency Test. Let 

«1 + U2 + U 3 + • • • + U n + • • • 

represent any infinite series of positive terms. Let r n represent the 
ratio of any term u n to the preceding term u n -\. Then 

W2 = Ufa, Uz = ^2*3, « 4 = W 4 , • • • u n = u n -\r ni • • • 

and consequently 

U\= Ui, 
U2 = Ufa, 

Uz = U2r 3 = Uir 2 r z , 
U4 = w 3 r 4 = Wir 2 r 3 r 4 , 



u n = u n -ir n = Uir 2 r 3 ri • • • r n , 



312 PLANE TRIGONOMETRY [chap, xv 

Adding Ui -f u* + u z + w 4 + • • • = Ui (i + r 2 + r 2 r 3 + r 2 r 3 r 4 + • • •) (i) 

<U!(i+R + R 2 + R*+ • • •) (2) 

> Ml (1 + r + r 2 + y* + • • •) (3) 

where R is greater than the greatest, and r less than the least, of 
all the ratios r 2 , r 3 , r 4 , . . . r n , . . . 

Now (2) is an infinite geometrical series which is convergent 
provided R is less than 1, and (3) is an infinite geometrical series which 
is divergent provided r is equal to or greater than 1, hence the 
infinite series 

«1 + ^2 + Uz + • • • + U n + • • • 

is convergent provided R is less than 1, divergent provided r is equal 
to or greater than 1. 

The ratio r n = — — is called the ratio of convergency or test ratio of 

U n -i 

the series Sw n . We have then the following theorems: 

A series is absolutely convergent if the test ratio is always less than 
some number R which is itself less than 1 . 

A series is divergent if the test ratio is always greater than some num- 
ber r which is itself equal to or greater than 1. 

Nothing is settled in case the test ratio is ultimately equal to 1. In 
this case other tests must be applied. 

These theorems remain true if, not from the first, but after some 
particular term, say the &th, the test ratio has the values stated. 
For the sum of k terms is finite, hence the whole series will be con- 
vergent or divergent, according as the series beginning with the &th 
term is convergent or divergent. 

The convergency test established in this article is known as the 
test-ratio test. It is sufficient for all the series treated in the re- 
maining portion of this chapter and the chapter following. In fact, 
the test-ratio test will answer most purposes of elementary mathe- 
matics; the cases in which the test fails, that is when the test ratio 
approaches 1 in the limit, form exceptional cases which can usually 
be avoided. There are a great many other convergency tests by 



i68] TRIGONOMETRIC SERIES 313 

which the convergency of a series can be settled in doubtful cases. 
The theory of series forms a separate subject of study, to the de- 
velopment of which many famous mathematicians have devoted 
their best efforts. 



168. Convergency of Special Series. 

(a) The exponential series* 

1+^ + ^7 + ^7+ • • • +^-+ • • • 
2! 3! nl 

where n ! (factorial n) stands f or 1 X 2 X 3 X ; • • X n. 

x n x n ~ 1 

Here u n+ i — — , u n = ; — , hence the test ratio 

n\ (n— i)l 

u n nl (n — i)l n 

If n is taken sufficiently large, - will become and remain less than 

n 

1 no matter how large x may be, provided only that it is finite; hence 
the series is absolutely convergent for every finite value of x. 

(b) The cosine series 

1 - + -+.. . ± — =F • • • 

2! 4! 6! (2wj! 

x 2n x 2n_2 
Here the general term is ± - — — , the preceding term =F > 

(2 n)l (2 n — 2)1 

hence the test ratio is 

± x 2n . =F * 2n ~ 2 — x 2 



(2n)l (2^—2)! 2n(2n—i) 

When n is taken sufficiently large, this ratio becomes and remains 
less than 1, therefore the series is absolutely convergent for all finite 
values of x. 

(c) The sine series 

/yO /yO /y* I /y2.ft -f- 1 

3! 5! 7! (2»+i)! 

* The reasons for the names given to the series in this article will appear 
shortly. 



314 PLANE TRIGONOMETRY [chap, xv 

In this case the test ratio is 

_[_ ^2n + l zn x 2n ~ l X 2 



(2W+l)! (2W-l)! 2«(2«+l) 

When n is taken sufficiently large this ratio becomes and remains 
less than unity, therefore the series is absolutely convergent for 
every finite value of x. 

(d) The logarithmic series 

/yL /y»o /y»4 /v* 

x — — | f- ••• ± — =F 

234 n 

x n x n ~ l 
u n = ± — , w n -i =I F , hence the test ratio is 

n n — 1 

±x n _^_ T x n ~ l __ — (n — 1) x 
n n — 1 n 

is a proper fraction which approaches iasw approaches 00 . 

n 

The test ratio will therefore approach a number less than 1, equal 
to 1, or greater than 1, according as x is less than 1, equal to 1, or 
greater than 1. 

The series is therefore absolutely convergent so long as x is less 
than 1. The test ratio fails to lead to any co elusion in the case 
x is equal to or greater than 1 . 

(e) The binomial series 

, , n (n — 1) o 1 n(n— i)(n— 2) o . 

1 + nx -\ x 2 H — x 3 + • • • 

2! 3! 

. n (n — i)(n — 2) . . . (n — r + 1) %r ■ 
r\ 
The general term is 

n(n — i)(n — 2) . . . (n — r -\- 1) 



r. 
r\ 



The preceding term is 



n (n — i)(n — 2) . . . (n — r -p- 2) r-1 
(r-i)! 



Dividing the general term by the preceding term and canceling the 
common factors, the test ratio reduces to 
n — r -\- 1 ^ _ 
r 



1 (n + 1 \ 



i68J TRIGONOMETRIC SERIES 315 

n ~\~ 1 
As r approaches 00, — ! — approaches o; the test ratio therefore 

r 

approaches -x as its limit, and we may conclude that the series 

is absolutely convergent so long as x is less than 1. 

Exercise 66 
Write down the first six terms of each of the following series: 

** n + 1 ** n l + 1 *~i nl 

Write down the general term of each of the following series: 

1-2 2-3 3.4 4.5 



X , X 



*v 1 «V 



6. — ^— + — h — h — V ' • ■ 

x + 1 # + 2 # + 3 x + 4 

1,1 1 1 1 

7. 1 1 1 ••• 

3 5 7 9 

/y» /y»£ /V*0 yV»4 ^y*0 



I I + £ 1 + 2 X 2 I + 3 X 3 I + 4 X 4 

Show that the following series are absolutely convergent: 

9 . I+ l 2 + 3_ 2 + 4 2 + . . . + jL + . . . 

^ 2 2 2 2 3 2 71 " 1 

, 2 • 2 , 3 • 2 2 , 4 • 2 3 , S ' 2 4 t 1 n » 2 n ~ 1 . 

10. I H + *— + ft — - + 2-j- + . . • H ^- + • • • 

3 3 3 3 3 * 

Ilm 2 2 1 3!* _|_ 4?£ 2 + s!^! _|_ ... i <^ + i) 2 * 71 " 1 1 
2! 3! 4! w! 

3# 2 , <x* 7 x Q , , / N„(2w+i)x 2n - 

12. 1 - ^_ + ^— - ^- + ... + (- i) TC * — -t—^ h • • • 

2! 4! 6! (2wj! 

Examine the following series as to convergency: 

13. 1+4+4+4+ • • • +4+ • • • 

2! 3! 4! nl 

14. I + - + - 2 + % + ' ' ' + - 2 + ' ' ' 

1 2 1 y n L 

Ans. Conv. f or x < 1. 

15. 1 + 2\x + $lx 2 + 4!^+ • • • +w!^ n_1 + • • • 

Ans. Div. 



316 PLANE TRIGONOMETRY [chap, xv 

16. i+± + ±.+±:+...+±-+... 

123 n 

Ans. Conv. f or x < 1, div. f or # > 1. 

17 . I_2 + I7_a_ 1 + 42_7I + ... 
2 6 12 20 30 42 

+ (-!)-! /*=! + _JL_\+... 

\ » n + 1/ 

^ — _ T 

(Suggestion. Find 6*2, 5 3 , 5 4 , .•■.., Sn= (— 1) From this 

n 

it is seen that the series is oscillating.) 

18. What is wrong with the following reasoning: 

Let 5'= 1 + 3+ 5+ 7 + 9+ ' ' ' 
S"= 2 + 4 + 6-1- 8+ 10+ • • • 

Adding S' + S"=i + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10+ ••• 

2 6 ,/ + 25 ,/ = 2 + 4 + 6 + 8 + io+ • • • = S", 
hence 2 5' + 5" = 6. 

19. Show that the series 

Ui — ih + u z — Ui + • • • + (— i) n_1 u n + • • • 

where u\ > ih > u% > w 4 > ■ ■ • > u nj and u n approaches o as n 
approaches 00 , is certainly convergent. 

169. The Number e. Consider the infinite series 

i+i + ~ + ^ + ^+---+-^+--- (1) 

I 2! 3! 4! n\ 

The test ratio is '- — = - > which is less than 1 for all 

n\ (n — ijl n 

values of n greater than 1, hence the series is convergent. 

Let the sum of the series be denoted by e. The sum of the first 
three terms alone is 2.5, hence e is certainly greater than 2.5. We 
will show that e is less than 2.75. 

1 1 



4! 3U S ] -3 
1 =— ^-<-^ 



51 3U-5 3*3 2 

— = " <-T L 7-> etc -> 

6! 3U-5-6 3^3 3 



i6 9 ] 










TRIGONOMETRIC 


SERIES 








hence 


























i + 


i 
4! 


+ 


1 
51 


+ 


z. + -- 


■<i 


■ + 


1 
3*3 


+ - 
3 


1 
!3 2 


+ - 

3 


1 

, 3 3 



3 X 7 



+ . . . 

The series on the right is a geometric series whose first term is — and 

3* 

whose ratio is - ; its sum by Article 166 is equal to 
3 

1 1 131 

T . = — . A = - = 0.25, 

3 ! i-f 3- 2 4 

therefore 1 1 1 h • ■ ■ < 0.2s, 

3! 4! S.I 6! T s ' 

and since 1 H 1 = 2.5, 

1 2! 

therefore * SSBI +I + -I+JL+JL+, •..+_*+... < 2 . 75 , 
1 2! 3! 4! »! 

that is 2.5 < e < 2.75. 

To determine e more exactly, we proceed as follows: Denote the 

terms of the series e by ui, u^, Uz, etc. 

U\ = 1. 000 000 00 

U2 = — = 1. 000 000 00 

1 

Uz = — = 0.500 000 00 
2 

U4 = B = 0.166 666 67 

3 

u$ = — = 0.041 666 67 
4 

u Q = - 5 = 0.008 333 33 

«7 = — = O.OOI 388 89 
W 8 = — = o.OOO I98 41 

7 

W9 = — = 0.000 024 80 
8 

W10 = — = 0.000 002 70 
Q 



318 PLANE TRIGONOMETRY [chap, xv 

2*10 o 

U\\ = = O.OOO OCX) 28 

10 

«12 = — " = 0.000 000 03 
II 

Adding 6*12 = 2.718 281 84. 

This is the sum of the first twelve terms of the series for e. There 
is an error in the last figure, owing to the neglected part of each of 
the decimal fractions added, but this error cannot exceed 10 X 0.5 
or 5 in the last decimal place. Besides this error there is the neglected 
portion of the series, which is u\z + Uu + Uy + • • 

Now 



U13 = 


I 

13! 










I 

14! 


< 




I 


Un — 


13 


1x3 


«15 = 


I 

TC' 


< 


T? 


I 
fir 2 



Adding u xz + Uu + Wis + •••<-^t(i+— + -^+---J 

13 ! \ 13 J 3 / 

11 1 



< 



i3!i- T V 12! 12 



From the computation above Un < 0.000 000 03, hence the neglected 

portion of e is less than — - < 0.000 000 003, that is, less than 3 in 

12 

the ninth decimal place. Therefore 

0=2.718 281 . . . correct to six places. 

170. The Exponential Series. By the binomial theorem 

/ , i\ n , 1 , n in — 1) 1 , n(n— i)(n— 2)1 , 

1 +- =1 +w-+-^— ■ - + — r ^-+ • • • 

\ nl n 2! n l 3! n 3 

= I+I+ v_^y + v — nil — »z + .... (I> 

2! 3! 

If n is greater than 1, each term of (1) is numerically equal to or 
less than the corresponding term of 

£=i + i + J :+ i r +--- (2) 

2! 3! 



170] TRIGONOMETRIC SERIES 319 

hence the series (1) is convergent for every value of n > 1. As w 
approaches 00 , each term after the second approaches as its limit 
the corresponding term of (2), hence 

the limit of ( 1 + - ) , as n approaches 00 , equals e = 2.718 • • • . (3) 
V nj 

(i\ nx 
1 + - ) by the binomial theorem. 
n) 

I - i\ nx , 1 , nx (nx — 1) 1 

\ n) n 2! n z 

, nx {nx — i )(nx — 2) 1 , 

~\~ — "T" * * * 

3! n? 



xlx ) x(x )(^~ 

V n/ , V n/\ 



= i+x+ * , AV +-^ ^ ^+ ■ • • . (4) 

2! 3! 

If n is greater than 1, each term of (4) is equal to or less than the 
corresponding term of the series 

I+* + ^+^f+ • • ', (5) 

2! 3! 

which in Article 168 was shown to be convergent for every value of 
x, hence (4) is convergent for every value of x. As n approaches 00 f 
each term of (4), beginning with the third, approaches as its limit 
the corresponding term of (5). Hence, as n approaches 00 f 

1 +i) =i+a;-h— + — +••• +- — h • • • (6) 
nj 2! 3! n\ 

Finally, by the law of exponents 

no matter how large n and x may be. As n approaches 00 f the 
expression on- the right approaches the series (6) as its limit, while 
the limit of the expression within the brackets on the left equals e, 
hence in the limit 

<r = 1 + x + ^ + ^ + • • • + % + • • • . (8) 

2! 3! n\ 

The series (8) is known as the exponential series because it is the 
equivalent of the exponential function e x . 



320 PLANE TRIGONOMETRY [chap, xv 

The exponential series may be used to compute the number cor- 
responding to any given natural logarithm. Assign to x any given 
value k, and let N be the number which is obtained by substituting k 
for x in the series (8). Then e h = N, and since e is the base of the 
natural system of logarithms, Article 36, we have by the definition of 

a logarithm, 

log e N = k, 

that is, N is the number which has k for its natural logarithm. 

171. The Logarithmic Series. Put i-\-y=e k , then &=log e (i+y), 

and (1 +'y) x =e kx = e xlo9 ^ l+y) . 

By (8), Article 170, 

(1 + y)» = e xl °e* d+») = 1 + xlog e (1 + y) + t* lo & (1 + y) ] 2 

2! 

+ [*log e (i + ;y)] 3 _|_ . . . 

By the binomial theorem 

(i+#=i+ot+ * (* ~ ^ y 2 + - (* ~ ^ ^ ~ 2 ^ y 3 + • • • 

2! 3! 

By the preceding article the first of these series is absolutely con- 
vergent, and so is the second provided y < 1, hence we may treat 
them like algebraic expressions containing a finite number of terms 
(Article 163). Equating the two series, subtracting 1 from each side 
and dividing out x, we obtain 

log. (1 + y) [1 + ff loge (1 +y)+^ log* 2 (i+y)+ ■ • -1 

= y + *—* f + (* ~ J ) (* ~ 2 ) /+.-'. 
2! 3! 

This equality holds for every value of #. As x approaches o, the 
series within the brackets on the left approaches 1, and the series on 

the right approaches y — 2- -\- 2C _ . . . ? hence in the limit 

2 3 

log e (i+2/) = 2/-^+^-^+ • • : +(-i) w - 1 ^+- • - (1) 
234 n 

Similarly, if y is negative, but numerically less than 1, we obtain 

loge(i-y) = -y-^-^- q ^- • • • - ^- • • • (2) 

234 ™ 



172] TRIGONOMETRIC SERIES 32 1 

The series (1) and (2), which differ only in the sign of y, are known 
as the logarithmic series because they are equivalent to the logarithms 
of 1 + y and 1 — y respectively. 

By the aid of (1) and (2) the natural logarithm of any given num- 
ber between o and 1 may be computed, but the actual computation 
of logarithms will be very much shortened by means of the method 
explained in the following article. 

172. Calculation of Logarithms. The series (1) and (2) of the 

preceding article have been shown to be absolutely convergent 

(Article 168, (d)); we may therefore subtract the second from the 
first and obtain 

log.(i+y)-lo & (i-y) = log.^±^= 2fy + ^ + ^ + ^+ • • ) (1) 

i-y V 3 5 7 / 



1 



If in this series we put y = , we obtain 

, n-\-i / 1 , 1 1 1.1 . 

lo & =2 — T~ + ~7 — Ii~V3 + "7 — T~^+r7 — TT^+ • • • 

n \2n-\-\ 3(2^+1)-* 5(2^+1) 7(2^+1) 



or 



loge (n + 1) = log e n + 2 ( — h 

\2 n + I 



3(2»+ i) : 



J 1 1 1 L . . .) ( 2 ) 

5 (2 ™ + i) 5 7 (2 ** + i) 7 I 

The series (2) is absolutely convergent so long as y is less than 
1, that is, so long as n is greater than o. If n is greater than 1 the 
series converges rapidly, so that but a few terms need be taken to 
obtain the first five or six decimal places of the number to which the 
series converges. The natural logarithm of any number may thus be 
readily computed provided we already know the logarithm of the 
next lower number. But the logarithm of 1 to any base is o (Article 
26), hence the logarithm of 2 may be computed. Knowing log e 2 
we may compute log e 3, thence log e 4, etc. Of course, only the 
logarithms of prime numbers need be computed by means of the 
series, for the logarithm of any composite number is equal to the 
sum of the logarithms of the factors of the number. 

To compute the natural logarithm of 2 we put in (2) n = 1, thus 

l0g e 2 = l0g e I+(- 2 +-^- 3 + - H - 5 +-^- 7 + • ' ' ) (3) 

\3 3'3 5V3 7*3 / 



322 



PLANE TRIGONOMETRY 



[chap. XV 



The actual calculation to five places of decimals may be conveniently 
arranged a follows. Denote the terms of the series in the parenthesis 
by «i, u 2 , m 3 , etc., then 



logei 



2.000 OOO O 



0.666 666 7 



0.074 074 1 



= 0.000 000 o 

1 = 0.666 666 7 = Mi 
3 = 0.024 691 4 = M2 



0.008 230 5 -T- 5 = 0.001 646 1 = M3 
0.000 914 5 -S- 7 = 0.000 130 6 = M4 

O.OOO IOI 5 -T- 9 = O.OOO Oil 3 = M5 

0.000 0113 V n = 0.000 001 = Uq 
13 



Adding 



0.000 001 3 
log e 1 + Mi + M2 + 



0.000 000 1 = M7 



+ u 7 = 0.693 x 47 2 



There is an error in the last figure, due to the neglected parts of 
the fractions added, but this error cannot exceed 7 X 0.5 or 4 in 
the last decimal place. Besides, there is the neglected portion of 
the series, consisting of the terms 



u 8 



M 9 = 



15-3 

2 



15 



Mi = 



I7'3 
2 



17 



19-3 



19 



<. 



< 



15*3 

2 
15*3 



15 



15 



Adding m 8 + m 9 + M10 + • • • < — —.J 1 + \ + - ■ + ■ ■ •) 

i5'3 15 V 3 3 / 

The series within the brackets on the right is a geometric series 

whose ratio is — = - . The sum of this series is = ° , hence the 

3 2 9 r_i a' 



neglected portion of the series is less than 



- * 8 
2 9 _ 2 



15*3 



15 



I20 • 3 



13 



From the computation above we already know that — = 0.000 001 3, 



13 



172] TRIGONOMETRIC SERIES 323 

hence = 0.000 000 01. The error, to which the above sum 

1 20 • 3 13 

0.693 147 2 is subject, may therefore cause a difference of at most 

1 in the sixth decimal place. It follows that log e 2 = 0.693 15 . . ., 

correct to 5 places of decimals. 

Similarly we obtain the following, each correct to 5 places of 

decimals : 

log e3 =log e 2 + 2 /l + -J- 5 + -JL^-i-+_L\= I>09 8 6l 
\5 3'5 3 5'5 5 7'5 7 9-57 

log e 4=2 l0g e 2 = I.386 29 

l0g e 5 =l0g e 4 + 2 ( - + -^— + — — + -5— ) = I.609 44 

\9 3'9 5*9 7 " 9 / 
log e 6 =log e 2+log e 3 = 1. 791 76 

l0g e 7 =l0g e 6 + 2 (— H — - H ^- ) = I.945 91 

\i3 3 • J 3 5 * 13 / 
log e 8 =3log e 2 = 2.07944 

l0g e 9 =2l0g e 3 = 2.I97 22 

l0g e IO = log e 2 + l0g e 5 = 2.302 59 

and so on. 

It will be observed that the number of terms of the series, which 
need be computed to obtain the logarithm correct to a given number 
of decimal places, grows smaller as n grows larger. When n is 43 
or more, the first term of the series suffices to give the first five 
places of the logarithm. 

When the natural logarithm of a number is known, the common 
logarithm is easily obtained from it, for by Article 35 

lo gl0 iV = i^. 
log e 10 

Now log e 10 was just found to be 2.302 59 ... , hence 

logio N = ^ = 0.434 29 . . . log e # (4) 

2.302 59 . . . 

The number 0.434 29 ... , more accurately 0.434 294 48 . . . , is 
called the modulus * of the common system of logarithms. We have 
then the following simple rule. 

* The base e, the modulus of the common system, and the natural logarithms 
of 2, 3 and 5, have been calculated to more than 250 places of decimals. 



3 2 4 



PLANE TRIGONOMETRY 



[chap. XV 



Rule : To find the common logarithm of a number, multiply the cor- 
responding natural logarithm by the modulus of the common system. 
Thus for the numbers from 2 to 10 inclusive we find the 





Common Logarithms 




logio 2 


= 0.434 294 . 


. X 0.693 T -S • • 


. = 0.301 03 ... . 


logio 3 


= 0.434 294 • 


. X 1.098 61 . . 


= 0.477 I2 • 




logio 4 


= 0.434 294 • 


. X 1.386 29 . . 


= 0.602 06 . 




logio 5 


= 0.434 294 • 


. X 1.60944 . . 


= 0.698 97 . 




logio 6 


= 0.434 294 . 


. X 1. 791 76 . . 


= 0.77815 . 




logio 7 


= 0.434 294 • 


• X 1.945 91 • • 


= 0.845 IO • 




logio 8 


= 0.434 294 . 


. X 2.079 44 • • 


= 0.903 09 . 




logio 9 


= 0.434 294 • 


. X 2.197 22 . . 


= 0.954 24 . 




logio 10 


= 0.434 294 . 


. X 2.302 59 . . 


= 1. 000 00 





each correct to five places of decimals. 

From (4) we have log e iV = 2.302 59 . . • logio N. 



(s) 



By means of (5) we can readily find any required natural logarithm 
from a table of common logarithms. It is therefore not very im- 
portant to have separate tables of natural logarithms. Thus, if at 
any time we needed to know the natural logarithm of 237.3 we would 
look up the common logarithm of 237.3 an( l multiply the result by 
2.30259 and obtain 

loge 237.3 = 2.30259 logio 237.3 = 2.30259 X 2.37530 = 5.46933 • • • 



173. Errors Resulting from the Use of Logarithms. In Article 
44 were given certain rules governing the accuracy to be expected 
in the results obtained by the use of logarithmic tables containing a 
certain number of decimals. Some of these rules we are now able to 
verify. 

Let N be any number obtained by logarithmic computation, and 
5 the error in the number due to the use of logarithms, so that the 
true result is N -\- 8, where of course 5 may be negative as well as 
positive. The difference between the logarithms of the true result 
and the result N is 

logio (N + 5) - logio N = logio ( —ff—) = lo gio f 1 + T^J 



.173] TRIGONOMETRIC SERIES 325 



/jlogfi + - j, where n = 0.43429 • • 
(By Article 172, (4) ) 



,8 8 2 , 8* 



N 2N 2 sN s 

(By Article 169, (1) ) 

= ii — > approximately, 
provided 5 is small as compared with N. 
This error ji — in the logarithm of N is the result of the errors in the 

logarithms as given in the tables. 

In a four-place table the value of a unit in the last place is 0.0001 = 

— > in a five-place table it is 0.00001 = — > in an w-place table the 
io 4 io 5 

value of the unit in the last place is The neglected part of 

io n 

any one logarithm does not exceed one-half of a unit in the last 

place; when several logarithms are added the positive and negative 

errors will tend to offset each other, but in special cases the errors 

may be accumulative. We will assume that the combined errors 

do not exceed a unit in the last place. This gives for an ^-place 

table 

logio (N + 8) - logio N = 11 — = -\i 

N io w 

from which 

g- N = 2.30259 j^ 

"^ IO n /A io n 

Putting n successively equal to 3, 4, 5, 6, 7, we find: 

In a 3-pl. table, 5 ^ 0.002 3 N, or less than J of 1 % of TV, 
In a 4-pl. table, 8 = 0.000 23 iV, or less than J of T V % of N, 
In a 5-pl. table, 5 = 0.000 023 N, or less than \ of T ^o % of N, 
In a 6-pl. table, 8 = 0.000 002 3 N, or less than J of toV % of -W, 
In a 7-pl. table, 8 = 0.000 000 23 N, or less than \ of Toio^ % of iV". 

It follows that the third figure of a number found from a three- 
place table, the fourth figure of a number found from a four-place 
table, the fifth figure of a number found from a five-place table, etc.,, 
cannot be relied upon with certainty. 



326 PLANE TRIGONOMETRY [chap, xv 

Exercise 67 

1. Compute the natural logarithms of 11, 101, 257, each to four 
places of decimals. Arts. 2.3979, 4.6151, 5.5491- 

2. From the results already obtained compute the natural logar- 
ithms of 12, 15, 0.05, J, each to four places of decimals. 

Ans. 2.4849, 2.7081, —2.9957, —1.0986. 

3. From the results of Problem 1 compute the common logarithms 
of 11, 101, 257, each to four places of decimals, and compare your 
results with those given in the table of common logarithms. 

4. By means of a table of common logarithms, find the natural 
logarithms of 7, 341, 0.0473. 

Ans. log e 34i = 5.4848, log e 0.0473 = ~ 3-°5 I 3- 

5. Prove that 

1 - 2 _]_ 4 . 6 , , 271 , 

« 3! 5! 7! (2»+l)! 

6. Compute - to five places of decimals, using the series in Prob- 

e 

lem 5. Ans. 0.36788 . . . 

7. Show that the compound amount A on P dollars for t years at 
r %, interest to be added to the principal as fast as it accrues, is - 

A = Pe rt . 
(Suggestion. Set up the expression for the compound amount 
when the period is the nth. part of a year and find the limit which this 
expression approaches as n approaches 00). 



Sum the series 1 -\ h - + • • ■ -\ h 

e e 2 e n 



Ans. 



e — 1 



174. Limiting Values of the Ratios SE^, ?HL£, x being the 

p radian measure of the angle. 

Let x be the radian measure of any angle less than 
ir/ 2. With the vertex O as a center and any radius 
OA — r describe an arc cutting the sides of the 
~c~~a angle in A and B. Join A and B. From B draw a 
Fig. 177. perpendicular to AO cutting AO in C. At A erect 
the perpendicular to AO cutting OB produced in T. Then 




*74l TRIGONOMETRIC SERIES 327 

CB = r sin x, arc AB = rx, AT = r tan x. 
OA X CB r 2 sin x 



The area of triangle OAB = 
The area of sector OAB = 
The area of triangle OA T = 



2 2 

OA X arc ,4.8 = r^x 

2 2 

OAXAT r 2 tanx 



2 2 

Also triangle OAB < sector OAB < triangle OAT, 

, , , . r 2 sin x . r 2 x . r 2 tan x 

that is, < — ■ < j 

222 

from which sin x < x < tan # (1) 

for every value of x less than - • 

2 

Dividing each term of (1) by the positive quantity sin#, 

1 < - — < > 



sin x cos x 
sin x ^ cos x 



, ,. Sill A, ^ L.US ^/ / \ 

hence 1 > > (2) 



x 

Now let x approach o, then cos x approaches 1, therefore , 

x 

which always lies between cos x and 1, must approach 1 also. 
Similarly, if we divide (1) by tan x 

cos x < — — < 1, (3) 

tan x 

from which it is seen that approaches 1 as x approaches o. 

x 

Summing up, — 

As x approaches o, the limit of equals 1. 

x 

As x approaches o, the limit of equals 1. 

x 

Corollary. 

As n approaches co , the limit of — ^-j- — - equals 1. 

x/n 

As n approaches 00 , the limit of \ / equals 1. 

x/n 



(4) 



(5) 



328 PLANE TRIGONOMETRY [chap, xv 

175. Limiting Value of cos"- and f^Ed^M. f as n approaches 00 . 

n \ sc/n J 

Put 

n 

y = cos n - = /i-sin 2 -Y, 
n \ ' n) 

then 

log y — - log f 1 — sin 2 - ) 
2 \ n) 

= — -(sin 2 - + -sin 4 - _i_ - sin 6 - +•••), by Art. 171, (2). 
2 \ n 2 n 3 » / 

The series in the parenthesis is less than 



Jv 1 • c* %X/ 



sin 2 — \- sin 4 — \- sin 6 — |- 
n n n 



which is a geometrical series, ratio sin 2 -, whose sum is (Art. 166) 

n 

sin 2 (x/n) _ sin 2 (x/n) _ 2 x 
1 — sin 2 (x/n) cos 2 (x/n) n 

hence, log y is numerically less than - tan 2 - = - tan , * 

2 n 2 n x/n 

Now let n approach 00 , then - approaches o, tan - approaches o, 

n n 

and , approaches 1 (Article 174, (5)), hence in the limit 

x/n 

log y approaches o, 

and consequently 

y — cos TC - approaches 1. 
n 

Also from Article 174, (2), 

^ sin (x/n) ^ x 
1 > — -r— 1 > cos - , 
x/n n 

and therefore 

^ /sin (x/n)\ n . n x 

1 > 1 — V^ 1 > cos " * 

\ x/n J n 

X IT r 

provided n is taken so large that - < - , that is, if n is taken suf- 

n 2 



176] TRIGONOMETRIC SERIES 329 

ficiently large ( ' - ) lies always between 1 and cos n -, but the 

\ x/n J n 

latter has just been shown to approach iasw approaches 00 , hence 

sin \x/ ) \ a j go approach r . Summing up: 
x/n ) 



As n approaches 00 , the limit of cos"— equalsjc: (1) 

ft 

As n approaches 00 , the limit of ( , ' } equals 1. (2) 

176. The Sine, Cosine and Tangent Series. In the equations 
(2) and (3), Article 159, namely, 

cosn$ = cos n - ^— ^cos n - 2 0-sin 2 

2! 

, n (n— i)(n— 2){n— %) n _ 4 Q . 4Q 
H * — — ^ cos 4 6 sm 4 6 — - • • 

sin n 6 = n cos 71 " 1 d sin 6 - n ^ n ~ T ^ n ~ 2>} cos n " 3 6 sin 3 

, n in — i)(n — 2){n — -\)(n — 4) n -^ a • la 

_j ^ L± LS Ql\ 3V cog w 5 fl sm 5 fl _ . . . 

5! 

putw0 = #, so that 0= -, then 

« X n \Yl ~~~ x ) n—o X • n X 

cos x = cos * J cos 2 - sm 2 - 

n 2! n n 

, W (w — l)(» — 2)(» — 3) n-4^ • 4# 

H — — — cos 4 - sm 4 • • • 

4! n n 

sm x = n cos 1 - sm — ' cos 3 - sm 3 - 

n n 3 ! n n 

, n (n — i)(n — 2)(n — x)(n — 4) n-s^ • *# 
_j v is n ai\ 32. cos ^ 5 _ sm 5 . . , 

5! n n 

An examination of the test ratio shows that both series are abso- 
lutely convergent for every value of n. As n approaches °° , 

cos n - approaches 1, 
n 



n , x sin (x/n) , 

__«-i_ i-z — l approaches x, 

n n n x/n 



w cos n ~ 1 - sin - = x cos 



330 PLANE TRIGONOMETRY [chap, xv 

n (n — i) n _ 2 x . 2 x I i\ x 2 n 2 x/sin (x/n)\ 2 x 2 

— * — - '- cos 2 - sin 2 - = i ) — cos 2 - v ' J ) appr. — , 

2! n n \ nj 2 ! n\ x/n ) 2 ! 

and similarly 

11 (n — 1) (n — 2) n o x . o x , x 3 

— — cos 6 — snr- approaches — > 

3! n n 3! 

w (w — 1) (n — 2) (n — 3) n a x . 4 x 1 x 4 

— — — — aL cos 4 - sm 4 - approaches — , etc., 

4! n n 4! 



Hence as n approaches 00 , the above series become 

sy* /"y*^ /-y*" /y»^* 

COS as = I . + _.-—. + _--... (1) 

2! 4! 6! 8! 

sina? = a? :+"? T + "T + * * * ( 2 ) 

3J sJ 7! 9! 

These series being absolutely convergent (Article 168, (b), (c) ), we 
may divide (2) by (1) and obtain 

tan x = x + ±-+±^- + i7iL _|_°?_*_ + _ # ^ 

3 15 315 2835 

The law of the exponents in each of the series (1), (2) and (3) is 
obvious, and so is the law of the coefficients in the series (1) and 
(2). The law of the coefficients in the series (3) is too complicated 
to be worked out by beginners. 

177. Computation of Natural Function Tables. We know that 
the functions of any angle whatever may be expressed in terms of 
the functions of an angle less than 45 . Moreover, the sine and 
cosine of any angle between 30 and 45 ° may be expressed in terms 
of the sine and cosine of angles less than 30 , for by Article 113, (1), 

, . . x 4- y x — v 
sm x -f- sm y = 2 sm — — — cos ■ — , 

2 2 

(1) 



. x + V • x — y 
cos x — cos y = — 2 sm — -^- sin • 

2 2 

If in (1) we put for x, 30 + 0, and for y, 30 — 6, and transpose the 
second terms on the left to the right-hand side of the equation, we 
obtain 

sin(3o°+0)= 2 sin3o°cos0— sin(3o°— 6)= cos#— sin (30 — 6), 
cos(30°+ d)= -2 sin3o o sin<9+cos(3o°-0)= -sin(9+cos (30°-^), ^ 



I77J TRIGONOMETRIC SERIES 33 1 

from which it appears that the functions of angles between 30 and 
45 may be obtained from the functions of angles less than 30 by 
giving to 6 successively the values from o° to 15 . Thus, if 6 = 5 , 

sin (30 + 6) = sin 35 = cos 5 - sin 25 , 
cos (30 + 0) = cos 35 = — sin 5 -f- cos 25 . 

A complete table of natural functions may therefore be readily con- 
structed provided we know the functions of angles between o° and 
30°. 

The sine and cosine of any angle less than 30 may be easily com- 
puted by means of the series in Article 176. 

Suppose we wish to compute the sine and cosine of io° correct to 
four places of decimals. 

The radian measure of io° is x = — = 0.174 53 . . . , and by means 

18 

of logarithms we find 

x 2 = 0.030 46 ... , x 3 = 0.005 3 2 • • • j 

x* = 0.000 93 ... , x 1 = 0.000 16 . . . 

Substituting these values in the series for the sine and cosine we 
have 

cos x sin x 

1. 000 00 x = 0.174 53 ... . 

= — 0.015 23 . . . ■ = — 0.000 89 . . . 

2! 3 ! 

x^ %? 
H = + 0.000 64 . . . H = + 0.000 00 . . . 

4! 5 J 



cos io°= 0.98481 . . . sinio°= 0.173 64 . . . 

In either case the error due to the neglected parts of the decimals 
added, cannot exceed a unit in the fifth place. In either case the 
error due to the neglected terms of the series is less than 

/y\) /y*i /yO yy\) 

^_ + ^ + ^l + ... < £_( I+x _|- X 2 +...) 

6! 7! 8! 6! 

The series in parenthesis is a geometric series whose ratio is x < 1. 

Its sum is , hence the neglected terms of either series add up to 

1 — x 

less than 



33 2 



PLANE TRIGONOMETRY 



x° 



(O.I74 53 ; • -) 6 , I 

6 ! 0.825 47 . . . 



[chap, xv 
= 0.000 000 05 . . . 



6 ! 1 — x 

In neither case, therefore, can the combined errors affect the fourth 
decimal place, and we have 

cos io° = 0.9848 . . . , sin io° = 0.1736 . . . , 

each correct to four places. 

As a check we have 

cos 2 io° + sin 2 io° = (0.9848) 2 + (0.1736) 2 = 1. 0000. 

It appears from the above computation that if the angle is less than 
io° the sine and cosine are given correct to four places by the 
formulas 



cos a? = 1 > 

2! 



sin <c = x — 



ar 



(3) 



These approximation formulas are sufficiently important to be 
remembered. 

We will next show how to compute a table of natural sines and 
cosines for intervals of i\ 

By means of the sine and cosine series we first compute the sine 
and cosine of i'. We find 



sin 1 = 0.000 290 555 2 . . . , cos 1 = 0.999 999 957 7 
If now we put in (1) x = 6 + i', y = 6 — i', we obtain 

sin (0 + i') = 2 sin 6 cos 1' — sin (6 — i'), 
cos {6 + 1') = — 2 sin 6 sin i' + cos (6 — 1'). 

Putting now 6 = 1' we find 



(4) 



sin 2 

COS 2 

Next put 0=2 
sin 3 
cos 3 



Similarly, if 6 = 3' 



sin 4 
cos 4 



= 2 sin 1 
= — 2 sin 1 

', then 
= 2 sin 2 
= — 2 sin 2 



= 2 sin 3 
= — 2 sin 3 



= 0.000 581 776 
= o-999 999 831 

= 0.000 872 665 
= 0.999 999 619 

= 0.001 163 553 



cos 1 — sin o 
sin i' + coso 

cos i' — sin 1 
sin 1' + cos 1 

cos i' — sin 2 

sin i' + cos 2' = 0.999 999 3 22 • • • > etc * 



178] TRIGONOMETRIC SERIES 333 

To construct a table of sines and cosines for intervals of 10", we 
should first compute the sine and cosine of 10" and then make use 
of the formulas 

sin (6 + 10") = 2 sin 6 cos 10" — sin (6 — 10"), 
cos (6 + 10") = - 2 sin sin 10" + cos (0 - 10"). 

178. Approximate Equality of Sine, Tangent and Radian 
Measure of very Small Angles. In Article 174 it was shown that 
the ratio of the sine to its angle expressed in radians, as well as the 
ratio of the tangent to its angle expressed in radians, approaches 1 as 
the angle approaches o. This means that for very small angles 
the sine, the tangent and the angle expressed in radians are approxi- 
mately equal. Thus, if we actually compute the sine and the tan- 
gent of 1" by means of the series in Article 176, and compare the 
results with the radian measure of 1", we shall find that the results 
agree to 15 places of decimals. The sine, tangent and radian measure 
of 1' agree to 11 places, and even when the angle is as large as i° the 
sine, tangent and radian measure are equal so far as the first five 
places of decimals are concerned. It follows that when the angle is 
small, say i° or less,* we may replace either or both the sine and 
the tangent by the angle expressed in radians, without affecting the 
first five places. 

Example i. Find the smallest value of x that will satisfy the 
equation 

2 sin x + 3 x = 0.0513. 

Solution. 

3 x = 0.0513 — 2 sin x 

< 0.0513, that is, x < 0.0171 (less than i°), 
hence we may replace sin x by x. Then 

2X + $X= $X = O.0513, 

x = 0.01026 radians = o°35 r 17". 

Note. The methods described in this chapter are not the 
methods that were used in calculating the tables now in use. The 
methods actually used were clumsy and laborious as compared with 
those we have studied. If the tables had to be calculated anew 

* In fact, the first five places are the same up to i° 59', that is, practically 2 . 



334 PLANE TRIGONOMETRY [chap, xv 

still more refined methods would be used, methods based on the 
calculus of finite differences, a branch of higher mathematics which 
cannot well be explained at this point. 

By means of the differential calculus the sine and cosine series 
can be much more easily derived than has been done in this chapter. 
In the differential calculus all the series of Article 168 and many 
others are derived by means of a single theorem known as Taylor's 
Theorem. 

Exercise 68 

i. Calculate the sine and cosine of 5 correct to five places. 

Ans. sin 5 = 0.08716, cos 5 = 0.9962. 

2. Using the results of problem 1, calculate the tangent, cotan- 
gent, secant and cosecant of 5 . 

Ans. sec 5 = 1.0038, esc 5 = 11.4737. 

3. By means of equations (4), Article 177, compute the sine and 
cosine of 5 . 

4. Compute the sine and cosine of 10" correct to 10 places. 

Ans. sin 10" = 0.000 048 481 4, cos 10" = 0.999 999 998 8. 

By means of the sine and cosine series verify the relations: 

5. sin 2 x + cos 2 x = 1 . 6. sin 2 x = 2 sin#cos#. 



7. Find the first three terms of the series for sec x. 






Ans. sec x = = 1 + — + ■* H * * • 

cos x 2 24 

8. An angle is to be corrected by an amount 8 which is known to 
be less than i° and is known to satisfy the equation 

28 = 1. 001 599 — cos 5 + sin 8. 
Find 8 expressed in minutes and seconds. Ans. 8 = 5' 30". 

9. A straight rail AB, 1 mile long, whose ex- 
tremities A and B are fixed, expands 1 inch, form- 
ing a curve ACB. Assuming the curve to be the 
arc of some circle, find the distance CD through Flg * I78 ' 
which the middle point of the rail moved during the expansion. 

Ans. 12 ft. 10.14 in. 




178] TRIGONOMETRIC SERIES " 335 

Suggestion. Let 

CD = x, AD = c, arc AC = c + 8, 



Then 



r = radius of circle of which AB forms an arc, 
6 = angle which AB subtends at the center. 



C ' a C + 5 n f o\ 

- = sin 6, = 9, x = r (1 — cos 6). 

Y T 



Eliminate r and solve for 6 on the assumption that 6 is very 
small, a natural assumption which may easily be verified. 



CHAPTER XVI 
HYPERBOLIC FUNCTIONS 

179. Series with Complex Terms. The trigonometric series 
which we considered in the last chapter had all their terms real. 
We have now occasion to consider series whose terms are not all 
real. 

Suppose that the terms of a series are complex numbers, the gen- 
eral term being of the form u n + iv n , then the sum of the first n 
terms may be written 

S n = (ui + ivi) + O2 + ivi) + (u 3 + iv 3 ) + • • • +(« B + iv n ) 
= U n + iV n , 

where 

U n = U X + u 2 + u 3 + • • • + u n , 

V n = vi + v 2 + v 3 + • • • + v n . 

If now, as w approaches 00 , U n approaches a finite limit U, and V n 
approaches a finite Umit V, then the series 

(wi"+ ivi) + («2 + ^2) + («3 + ^3) + • • • + («» + iv n ) + • • •, 

it is said to be convergent, and by its sum S is understood the expres- 
sion 

S=U + iV. 

It follows that in order that a series with complex terms may be 
convergent it is both necessary and sufficient that the real and the 
imaginary parts of the series are separately convergent. 

If either the real part or the imaginary part of a series is divergent 
the series is considered divergent. 

If the general term u n + iv n be expressed in the trigonometric 
form r n (cos 8 n + i sin 6 n ), then 

S n = ri (cos 6i + i sin 61) + r 2 (cos 6 2 + i sin 2 ) 

+ 7*3 (cos 63 + i sin 3 ) + • • ■ + r n (cos 6 n + * sin d n ) 

= ^n+*T n , 

336 



180] HYPERBOLIC FUNCTIONS 337 

where 

U n = ri cos di + y% cos 62 ,+ r 3 cos 3 + • ■ ■ + r n cos n , 
F n = ri sin X + r 2 sin 2 + r 3 sin 3 + • • • + r n sin 6 n . 

Each sine and cosine is less than or at most equal to 1, hence 

U n ~r 1 + r 2 + r 3 + • • • + r n ,' 
F n = /1 + r 2 + r 3 + • • • + r n , 

and consequently, as n approaches go , U n and F n will each approach 
a finite limit, provided the infinite series 

ri + r 2 + r 3 + * • • +r n + • • • 

is convergent, that is, 

If the moduli of the terms of a series with complex terms form a con- 
vergent series, the series is convergent. Since every modulus is posi- 
tive, the series will in this case be absolutely convergent. 

It may be shown that series with complex terms which are abso- 
lutely convergent may be added and subtracted, multiplied and 
divided, by the same rules as polynomials, but this is not true of 
conditionally convergent series, that is, of convergent series which 
are not absolutely convergent. 

180. Definition of the Trigonometric Functions of Complex 
Numbers. 

If in the exponential series 

c*=i + * + 3L + ~+ • • • +*r +■'•■• W 

2! 3! n\ 

we place x by z = x + iy = r (cos + i sin 0), the series of the 
moduli is 

r 2 r 3 r n 

i + r+ r -+ r -+ ■ ■ ■ + r -+ : ■ ; (1') 

2! 3! n\ 

which we know to be convergent for every value of r (Article 168, (a)). 
Hence by Article 179 the series is absolutely convergent. Its sum 
may conveniently be denoted by e z = e x + iy = e r ^ cosd + * sin ^) ? that is, 

e*=i + z +4+4+-- • • + ? 7+ '• • • W 

2! 3! n\ 



338 PLANE TRIGONOMETRY [chap, xvi 

Similarly we may define cos z and sin z by the series 

Z 2 , Z 4 Z 6 . / x 

cos z = i r • • • (3) 

2! 4! 6! 

2 3 1 2 5 Z 7 , / \ 

sin 2 = z + — + • • •, (4) 

3| 5! 7! 

for in each case the series of the moduli consisting of alternate terms 
of the series (i') is convergent. 

The remaining trigonometric functions of complex numbers are 
defined by means of the relations 

sinz .• 1 1 1 / n 
tanz= , cotz = , cscz = - — , secz= (5) 

cos z tan z sin z cos z 

With (2) as a definition of e z , it may be shown that 

e Z l . e Z 2 — g*i + *2 ? an( i (g 2 l) 2 2 = e ^2 f 

that is, we may treat e 2 , where z is a complex number, exactly as we 
treat e x , where x is a real number. 

Similarly, with (3), (4) and (5) as definitions of the trigonometric 
functions all the familiar fundamental laws may be shown to hold 
when z is a complex number as well as when x is a real number, z of 
course is a number and can be represented by an angle only when it 
is real. 

181. Euler's Theorem, e iQ = cosQ + * sw6. 
First Proof. If in (2), Article 180, we put z = id, 



4* = 1 + i$ + t0 + 

2! 


i 3 d 3 , ; 4 4 . i h B h . 
3! 4! 5! 


( 1 ° 2 + ° 4 
\ ~ 2! 4!" 


-■M-t + ?, 


= cos 6 + i sin 6. 





(1) 



This result, known as Euler's theorem, is next to DeMoivre's 
theorem one of the most important results of modern analysis. 
Since every number real or complex can be expressed in the form 
r (cos 6 + i sin 6), the theorem may be stated thus: 

Every number .may be expressed in the form re id , where r is the 
modulus and 6 the argument of the number in its trigonometric form. 



182] 



HYPERBOLIC FUNCTIONS 



339 



Second Proof. Euler's theorem may be established by the aid of 
DeMoivre's theorem as follows. By DeMoivre's theorem 

/ ft a \ n 

cos 6 + i sin 6 — j cos — |- i ^ sin - ) , for every value of n. 
\ n n) 

a 

Consider 6 constant, and let n approach oo . Then sin — approaches 

n 

-, cos— approaches i, that is, 
n n 

(cos — (- * sin — ) approaches the same hmit as ( i + — ) • 
\ n n) \ n J 



Put— = — , that is, n = mi$, then 
n m 



n j \ m) 



i + 



m 



YTttd 



As n approaches oo , m also approaches oo , but 

the limit of (i H ) , as m approaches oo, is e (Article 170, (3)), 



m) 



hence 



id 



the Hmit of f 1 -\ ] , as n approaches 00 , is e id , and we have 

cos 8 + i sin 6 = e^. 



182. Geometrical Representation of Euler's Theorem. That 

(i6\ n 
1 -\ — ) , as n approaches 00 , is cos 6 + i sin 6 may be 
n) 

seen from the following geometrical construction. 

Let AOQ be any angle d, AQ the subtended 

arc drawn with a unit radius OA = 1. At A 

construct AP = arc AQ = 6 perpendicular to 

OA, and let A Pi be one of the n equal parts of 

X AP. Then AP X = L 
Fig. i79- n 

Now consider OX the axis of reals and OY the axis of imaginaries, 

then the directed line OP\ represents the expression 

OA + i-AP!= i + -- 

n 




340 PLANE TRIGONOMETRY [chap, xvi 

Construct in succession the similar right triangles OP1P2, OP^Pz, 
. . . , OP 3 P n , n in number including the triangle AOP h each having 
for its base the hypotenuse of the triangle immediately preceding, 
then from the proportionality of the homologous sides we have 



OP 2 : OPi = OPi : OA , or OP 2 = OPi (since OA = 1), 
OP 3 : OP 2 = OP 2 : OP h or OP z = OP?. 



id\ n 



Similarly OP n = OPi. 

id 
The directed line OP\ represents 1 -\ — 1 

n 

I iB\ 2 
hence the directed line OP 2 represents f 1 -\ — ) 1 

\ nj 

and the directed line OP3 represents ( 1 -\ ) 

V n) 

('p\ 
1 H — ) 
n) 

Now let n be indefinitely increased, A Pi becoming correspond- 
ingly smaller. As n approaches 00, the straight lines A P h P1P2, 
P2P3, . • . approach equality, the broken line AP1P2P3P71 approaches 
the arc AQ, and the directed line OP n approaches the directed line 
OQ. But the directed line OQ represents OB + i • BQ = cos 6 + 

(if)\ n 
1 -\ — ) , as n 
n) 

approaches 00 , is cos 6 -f- i sin 6. 

183. Exponential Form of the Sine and Cosine. If in 

£^=cos0 + isin0 (1) 

we replace 6 by — 6, we get 

e -iQ — CQS (_ fl) _|_ I gm (_ 0) _ CQS Q — I s i n ? (2) 

and solving (1) and (2) for cos 6 and sin 6 respectively we have 



= ! , sm = 



e iQ — e~ 



co S e = " ■ p , sin6=^ ^-. (3) 

2 2 i 

If in (1) we put 6 successively equal to -, tz and 2 t, we obtain the 

2 

important results 

in 

e 2 = i, e in = — 1, e 2in = 1, (4) 



183J HYPERBOLIC FUNCTIONS 341 

that is, to increase the exponent of e by — is to multiply the expres- 

2 

sion by i; to increase the exponent of e by iw is to change the sign of 
the expression ; to increase the exponent of e by 2 iw is to leave the 
expression unchanged. 

The relations (3) can be shown to hold for any value of 6 whether 
real or complex. In fact, assuming a knowledge of imaginaries and 
of exponentials (expressions of the form e z ), we might have defined 
the sine and cosine by means of these relations. With these rela- 
tions as a basis, the whole of trigonometry becomes an easy applica- 
tion of algebra. We will show by some examples how the formulas 
of trigonometry could be derived from the relations (3). 

Example i. To prove that sin 2 6 = 2 sin 6 cos 8. 
Solution. 

e 2id e —2id e i0 g-id gid _|_ g— id 

sin 2 6 = = 2 ; — • = 2 sin 6 cos 6. 

2i - 2% 2 

Example 2. To prove that cos ( - — 6 J= sin 6. 
Solution. 

'ir A_ e 2 +e 2 % e ie —e~ l0 



cos [- - = ; = — = - "-— = sin 0. 

,2 / 2 2 2 1 

Example 3. To prove that 2 sin d cos = sin (0+<£) + sin (d—<f>). 
Solution. 

2 sin 6 cos 9=2 : — . - — !— - — 

2 i 2 

e i(o+<j>) _ e -i(d+<f>) _|_ e i(d-<j>) _ e _t(0-<£) 
2 i 
= sin (0 + 0) + sin (0- 0). 

Exercise 69 

By means of the relations (3) and (4), Article 183, prove the follow- 
ing: 

1. cos 2 6 + sin 2 d=i. 2. cos 2 6 — sin 2 = cos 2 0. 

3. coso = 1, sino = o. 

4. e e+2ni * = e*, where w is any integer. 



34 2 PLANE TRIGONOMETRY [chap, xvi 

5. cos (2 n-w + 6) — cos0. 6. sin ( - — J = cos 0. 

7. cos (jr — d) = — cos0. 8. 2 cos cos (j> = cos (0+0)+ cos (0— <£>). 

4 /I + COS 2 Q . /i — COS 2 . a 

9. V/ — = cos 0, 1/ = sin 0. 

10. By actually multiplying the series for e u and e v show that 

e u . e v = e u+v . 

in —in 

11. Show that e 3 + e 3 =1. 

12. Show that cos id and i sin z'0 are real. 

13. Prove Euler's theorem, 

e iz — cos z _|_ I gm ^^ 

where 2 is any complex number. 

14. Prove DeMoivre's theorem 

(cos z + i sin z) n = cos nz + i sin nz, 
where z is any complex number, assuming the result of Problem 13. 

184. Hyperbolic Functions Defined. Euler's theorem, 

cos0 + z'sin0 = e id , (1) 

is not limited to real values of 0, but holds true when is imaginary 
or complex, for if we put for cos 0, sin and e %e the series which 
define these functions for complex values of 0, the two sides of (1) 
become identically equal. We may therefore replace by id and 
again by — id and obtain 

cos id + i sin id = e~ , 
cos id — i sin id = e +e . 

Half the sum and difference give respectively 

cos *9 = — i ) — * sin it) = • 

2 2 

For reasons which will appear presently these expressions are 
known as the hyperbolic cosine of and the hyperbolic sine of re- 
spectively. The hyperbolic tangent is defined as the ratio of the 
hyperbolic sine to the hyperbolic cosine. The reciprocals of the 
hyperbolic cosine, sine and tangent are called the hyperbolic secant, 



185] HYPERBOLIC FUNCTIONS 343 



cosecant and cotangent respectively. Their most common abbre- 
viations are cosh,* sinh, tanh, sech, csch, coth. 
We have accordingly 

cosh 8 = e ~^ e = cos i8, 



e^—e 



sinh = = — i sin i8, (2) 

2 

tanh8= e [~ e ~\ [=- itani8, 
e Q -\-e~v 

and the reciprocal expressions for sech 6, csch 6 and coth 6. Notice 
that so long as 6 is real, e e and e~ d are real and each of the hyperbolic 
functions is real. Their graphs have already been drawn. (Exercise 
61, Problem 8, and Fig. 172.) 

185. Duality Between the Circular and Hyperbolic Functions. 

To every property of or relation between the trigonometric or 
circular functions there exists a corresponding property of or rela- 
tion between hyperbolic functions.! The formulas expressing these 
properties and relations could all be derived from the expressions of 
the hyperbolic functions in terms of exponentials (Article 184, (2)), 
but an easier way to discover them is to substitute the values 

cos id = cosh 6, sin id = i sinh 6, tan id = i tanh 6, (1) 

obtained from Article 184, (2), in the corresponding formulas for the 
circular functions. We shall illustrate both methods in the examples 
which follow. 

Example i. What relation between hyperbolic functions cor- 
responds to the relation cos 2 d + sin 2 =1? 

Solution. Put for 6, id, then cos 2 id -f- sin 2 id = 1. Substituting 
for cos id and sin id their values from (1) we have 

cosh 2 + f 2 sinh 2 0= 1, 
or cosh 2 — sinh 2 # = 1, 

which is the required relation. 

* Pronounced cosh, shin, than, shec, etc., in academic slang. Some writers 
use the abbreviations h-cos or hycos, h-sin or hysin, etc. 

f Both circular and hyperbolic functions are in fact members of a more gen- 
eral class, the modo-cyclic functions, defined by the equations 

sin* 6 = - (e d/k - e~ 9/k ) , cos* 6 = § (e e/k + e ~ e/k ) , tan& d = sin* 0/cosfc 6, etc. 



344 PLANE TRIGONOMETRY [chap, xvi 

If we had been asked to prove the relation cosh 2 — sinh 2 6 =i, 
we might have proceeded as follows : 

/ P e _i_ p -e\2 p 2d I _i_ p -26 
By Article 184, (2), cosh 2 - '- ] -i - 



Subtracting, 

cosh 2 — sinh 2 = 1. 

Example 2. What property of hyperbolic functions corresponds 
to the property sin (0 ± 2 mr) = sin ? 

Solution. Replace by id, then 

sin ($ ± 2 W7r) = sin [i(6 T 2 nm)\ = sin $. 
Introducing the relations (1) 

i sinh (0 =F 2 ?mr) = * sinh 0, 
that is, sinh (^2 wwr) == sinh 0. 

This shows that the hyperbolic sine has the imaginary period 
2 iir just as the circular sine has the real period 2 ir. Similarly it 
may be shown that each of the hyperbolic functions has the period 
2 iir. The hyperbolic tangent and cotangent have the smaller 
period iir. 

If we had been asked to verify the relation sinh (0 T 2 niir) = sinh 0, 
we might have proceeded thus : 

g0=F2niV e ~ 0±2mV J& „-Q 

sinh (0 T 2 niir) = = = sinh 0, 

2 2 

for according to Article 183, (4), e ±2ni * = 1. 

Example 3. What formula for hyperbolic functions corresponds 
to the formula 

sin + sin 9 = 2 sin — —^ cos r ? 

2 2 

Solution. Replace by id, and by i<£, we have 

. •/, 1 • • / .iB-\-id> id — id> 
sin w + sin £9 = 2 sin — cos — , 



186] HYPERBOLIC FUNCTIONS 345 

whence by (1) 

i sinh + i sinh 6—2i sinh -^-?- cosh 2 . 

2 2 

Dividing by i 

sinh + sinh <£ = 2 sinh 2 cosn Z 9 , 

2 2 

2 tanh 



Example 4. Prove the relation 

1 + tanh 2 

Solution. By Article 184, (2), 
tanh = -— r 1 + tanh 2 = 1 + ' 



/ e d -e- e \ 2 = 
\e d + e- e ) '" 



e e +e- e \e e +e- e l (e e +e- e ) 2 ' 

hence 

2 tanh = 2 (e e - e~ 6 ) # (e d + e- e ) 2 
i + tanh 2 e d + e- 2(e 2 * + e- 2 *) 

_ (e e - e~ d ) (e d + e~ d ) _ = e 2g - e~ 29 
^Q+e-zo \ e 2d^. e -2d 

= tanh 2 0. 

Below we give for purposes of reference and comparison a table 
of the principal formulas for the circular functions and in parallel 
column the corresponding formulas for the hyperbolic functions. 

186. Table of Formulas. 

Circular Functions. Hyperbolic Functions. 

1. sin • esc = 1. sinh u • csch u = 1. 

2. cos • sec = i. cosh u • sech u = 1. 

3. tan 0« cot 0=i. tanh w cothw = 1. 

4. cos 2 + sin 2 0=i. cosh 2 u — sinh 2 u— 1. 

5. 1 + tan 2 = sec 2 0. 1 — tanh 2 w = sech 2 w. 

6. cot 2 + 1 = esc 2 0. coth 2 u — 1 = csch 2 u. 

7. sin = 0. sin - = 1. sinho = o. sinh (± °o) = °°. 



8. cos o = 1. cos - = o. cosh 0=1. cosh (± 00)= ±00. 



9. tan = 0. tan - = 00. tanho= o. tanh (±00)= ±1. 
2 



IO. 



II. COS 



346 PLANE TRIGONOMETRY [chap, xvi 

>. sin I - ± 8 J = cos 0. sinh I— ± u\= i cosh u. 

I- ± Oj = T sin 0. cosh (— ± u\= ± i sinh u. 

12. tan(- ± 0) = ± cot0. tanh ( — ± u\= ±coth u. 

13. sin (r ± 0) = =F sin 0. sinh (wr ± w) = =F sinh w. 

14. cos (ir ± 0) = — cos 0. cosh (wr ± u) — — cosh u. 

15. tan (t ± 0) = ± tan 0. tanh (iir ±m)=± tanh w. 

16. sin (2 7r + 0) = sin 0. sinh (2 wr + w) = sinh w. 

17. cos (2 7r + 0) = cos 0. cosh (2 iV + w) = cosh w. 

18. tan (2 7r + o) = tan 0. tanh (2 «r + w) = tanh u. 

19. sin (—0) = — sin 0. sinh (— w) = — sinh u. 

20. cos (— 0) = cos 0. cosh (— u) = cosh u. 

21. tan (— 0) = — tan 0. tanh (— . u) = — tanh u. 

22. sin (0 ± </>) = sin cos <j> sinh (w ± ») = sinh u cosh z> 
±cos0sin0. . ± cosh w sinh v. 

23. cos (d ± cf>) = cos cos cosh (« ± a) = cosh w cosh v 
=F sin sin cj). ± sinh w sinh v. 

, n . ,\ tan ± tan c6 ,, . >. tanh w± tanh u 

24. tan(0±9) = -^ tanh (w ± z>) = — • 

1 =F tan tan 9 1 ± tanh u tanh z> 

25. sin 2 = 2 sin cos 0. sinh 2u = 2 sinh w cosh w. 

26. cos 2 = cos 2 — sin 2 cosh 2 u = cosh 2 w + sinh 2 u 
= 2 cos 2 — 1 =2 cosh 2 w — 1 
= 1 — 2 sin 2 0. =1 + 2 sinh 2 u. 

n 2 tan .i 2 tanh w 

27. tan 26 = tanh 2u = 



tan 2 6 1 + tanh 2 w 



28. sin 3 = 3 sin — 4 sin 3 0. sinh 3^ = 3 sinh w + 4 sinh 3 w. 

29. cos 30 = 4 cos 3 — 3 cos 0. cosh 3^ = 4 cosh 3 w — 3 cosh u. 

30. sin + sin <f> sinh w + sinh v 

. + <£ — d> . , w+fl 1 w— z> 
= 2 sm — —- i - cos — • = 2 sinh cosh • 



186] HYPERBOLIC FUNCTIONS 347 

31. sin — sin cj> sinh u — sinh v 

6-\-<f> ■ 6 — (f> 1 u-\-v • i w — z> 
= 2 cos — — - sin • = 2 cosn sinn • 

22 22 

32. cos + cos </> cosh w + cosh v 

d-\-d> 6—<f> 1 u-\-v , w— y 

= 2 cos — — - cos — - • = 2 cosn cosh • 

22 22 

33. cos — cos (f> cosh w — cosh v 

. -\-d> . 6 —(f) • 1 u-\-v . ,u — v 
= — 2 sin — — sin • = 2 smh — — sinh . 



4 /i + cos0 1 u /coshw+i 

34 . cos _ =v /___ C osh-= V / 

.0 ,/i- cos . , w /cosh w — 1 

35. sm-=W sinh-=i/ 

2 ▼ 2 2 T 2 

, ,. sin0 ■• sinhw 

36. 11m = 1. nm = 1. 

0=0 u=o u 

,. tan0 r tanhw 

37. 11m = 1. 11m = 1. 

0=0 u=0 U 

38 . lim / rin(g/») Y = , lim ( sinh (uln) Y = , 

39 . lim / tanjgMy = , lim / tanh( M /») y _ , 

40. sin = 1 • • • smh u = u-\ H + 



3! 5! 3! 5 

41. cos 6=1 H -— • • • coshw = H H ■ + • 

2! 4! 2! 4! 

42. (cos ± i sin 0) n (cosh u ± sinh w) n 

= cos «0 ± i sin w0. = cosh ww ± sinh nu. 

43. cos0 ±2sin0 = e ±id . cosh w ± sinhw = e ±M . 

44. sin = ; sinh u = 



21 2 



e iO+ e -id e u _J_ e -u 

45. cos0 = coshw = 



348 PLANE TRIGONOMETRY [chap, xvi 

46. i sin 6 = sinh id. i sinh u = sin iu. 

47. cos 6 = cosh id. cosh w = cos iu. 

48. 2 tan 6 = tanh #. i tanh « = tan iu. 

Exercise 70 

1. Prove the formulas (7), (8) and (9) for hyperbolic functions. 

2. Verify formula (10) for hyperbolic functions. 

3. Verify formula (14) for hyperbolic functions. 

4. Verify formula (18) for hyperbolic functions. 

5. Verify formula (21) for hyperbolic functions. 

6. Verify formula (27,) for hyperbolic functions. 

7. Derive formula (26) for hyperbolic functions. 

8. Derive formula (36) for hyperbolic functions. 

9. Derive formula (42) for hyperbolic functions. 

10. If y = cosh x, show that x = log (y ± v y 2 — 1). 

11. Ifv = tanh^, show that x=-\ogl *-\. 

2 \i-yj 

187. The Inverse Hyperbolic Functions. 

Let y = cosh x = > (1) 

2 

then x = cosh -1 v is called the inverse hyperbolic cosine 3/. 

Put e x = z, then e~ x = -, and we have from (1) 

z 

z -+■ - = 2 y, or z 2 — 2 yz + 1 = o, 
z 

from which, on solving for z, 



e x = z = y ± V ^ 2 — 1, 

and taking the logarithm of both sides of this equation, 

a? = cosh" * ?/ = log (y ± Vy 2 — 1). (2) 



188] HYPERBOLIC FUNCTIONS 349 

In like manner from y = sinh x = , (3) 

2 

x = sinh -1 y, the inverse hyperbolic sine y. 
Putting in (3) e x = z, we obtain 

z = 2 y, or z l — 2 yz — 1 = o. 

z 



Solving for z, e x = z = y ± \/y 2 + 1. 

The minus sign cannot be used, for e x is positive for every value of 

x, while y — *vy 2 + 1 is negative, V y 2 + 1 being greater than v. 
Hence, on taking the logarithm of both sides of the above equation, 



x 



= sinh" x y = log (y + v 7 ^ 2 +1). (4) 



If y = tanh x = g ■ _ g ? then x = tanh -1 y, the inverse hyperbolic 

tangent y. 

Again, we put e x = z and solve the resulting equation for z, 



» i — y 

T I i - V 

from which x = tanh -1 y = -log z , (5) 

2 1 — ;y 

In the same manner we obtain for the remaining inverse hyper- 
bolic functions 

<c = coth~ 1 y = Ilog^±J.. (6) 



y- 1 



oc = sech - x y = log i-= - 



2/ 2 



u-i 1 I + Vi + 7/ 2 
a? = csch x y = log — ! L -^- • 

V 



(7) 
(8) 



188. Geometrical Representation of Hyperbolic Functions. 

We will now show that the hyperbolic functions may be expressed as 
ratios of certain lines connected with the equilateral hyperbola, just 
as the circular functions are expressed as ratios of certain lines con- 
nected with the circle. These relations will be best understood by 



35° 



PLANE TRIGONOMETRY 



[chap. XVI 



developing the corresponding results for the circle and hyperbola in 
parallel columns. 





Fig. 1 80. 



Fig. 181. 



Let x = a cos 0, y = a sin 0, Let x = a cosh u, y = a sinh u, 

then since cos 2 + sin 2 = 1 , then since cosh 2 u — sinh 2 u = 1 , 
x 2 -|- >> 2 = a 2 (cos 2 + sin 2 0) = a 2 , x 2 — y 2 = a 2 (cosh 2 u — sinh 2 u) = a 2 ,, 
that is, x and y are the coordi- that is, x and y are the coordi- 
nates of a point P on a circle nates of a point P on a hyperbola 



whose radius is a. 



whose semi-major axis is a. 



COS0 = 


x_OB 
a OA' 




u x OB 
cosh u = - = — , 

a OA 




sin 6 = 


y_BP 
a OA 1 




. , y BP 
smh u = - = > 

a OA 




tan0 = 


sin 6 BP 
cos 6 OB 


- AT . 
' OA 


, sinh w 5P 

tanh w = — - — = 

cosh u OB 


AT 
OA 



If OA is taken for the unit of 
measure, 

OB represents cos 0, 

BP represents sin 0, 

A T represents tan 0. 



If OA is taken for the unit of 
measure, 

OB represents cosh u, 
BP represents sinh u, 
A T represents tanh u. 



It is plain, now, that the hyperbolic functions are related to the 
hyperbola in the same way that the circular functions are related to 
the circle. For this reason the functions are named "hyperbolic 
functions." 

189. The Area of a Hyperbolic Sector. In Article goa it was 

a 2 d 
shown that the area of a circular sector is equal to — . We will now 

2 

derive this result in another way, and show in parallel column that 

m like manner the area of a hyperbolic sector is equal to 

2 



i8 9 ] 



HYPERBOLIC FUNCTIONS 



351 





Fig. 182. 

Let P be any point on the circle 
corresponding to and 



Fig. 183. 

Let P be any point on the hy- 
perbola corresponding to u and 



a, Pt, 



Pk, P k + h 



Pi, P* 



Pk,P 



k + h 



points on the circle corresponding 
to 



20 
~> : 

n n 



k0 (k+i)0 

- > : 

n 



n 



respectively, n being any integer 
arbitrarily chosen. 

Let the coordinates of Pi, P2, 
etc., be xi, yi\ X2, y 2 ; etc., re- 
spectively. 

The area of triangle OP k P k +i 

= area of triangle OCPk+i 
+ area of trapezoid CP k +iP k B 
— area of triangle OBP k , 



points on the hyperbola corre- 
sponding to 

u 2 u ku (k + 1) u 

— , — , . . . — , ^ '. — , . . . 

n n n n 

respectively, n being any integer 
arbitrarily chosen. 

Let the coordinates of P h P 2 , 
etc., be xi, 3/1; x 2 , y 2 ; etc., re- 
spectively. 

The area of triangle OP k P k +i 

= area of triangle OCP k +i 

— area of trapezoid CP k +\P k B 

— area of triangle OBP k , 



_ Xk+iVk+i 
2 


_ %k+iyk+i 
2 


1 (x k — x k+1 )(y k + y k +i) 
2 


(xk+i— x k )(y k +y k +i) 

2 


2 


x k y k . 
2 


Simplifying, 


Simplifying, 


area of triangle OP k P k +i 

_ (yk+i%k—%k+iyk) 
2 


area of triangle OP k P k +i 

_ (yk+iXk—x k+1 y k ) 
2 



352 



PLANE TRIGONOMETRY 



[chap. XVI 



a 2 V 



(k+~i)6 kd 

sin - — ! — — cos — 

n n 

(k + i)0 . kd 

cos - — ! — — sin — 
n n 



a 2 . r(*+i)0 

= — sin - — 
2 L 



n 



= —sin — ' 
2 n 



n _ 



= — sinn- — ! — — cosh — 
2 |_ n n 

i (k + i)w . -, foT| 
— cosh - — ! — — smh — u 

n n \ 

= ?L 2 sinh r(i±iV_M 

2 n n\ 



a L . i u 
= —smh -• 

2 n 



(i) 



Since (i) is the area of any one Since (i) is the area of any 

of the triangles which make up the one of the triangles which make 

polygon up the polygon 

OAP& . . . PkPjc+i • • • P, OAPtPz . . . P k P k+1 . . . P, 

the entire area of the polygon the entire area of the polygon 



a- 



6 



= n— • sin — 
2 n 

_ a?B m sin {Bin) 
2 B In 



a* . i u 
— n— • smh — 

2 n 

_ ahi ^ sinh fa/w) 

2 «/» 



(2) 



Now let n be indefinitely in- Now let n be indefinitely in- 
creased, then the area of the creased, then the area of the 
polygon approaches as its limit polygon approaches as its limit 
the area of the circular sector the area of the hyperbolic sector 
OAP, which we will denote by S c , OAP, which we will denote by Sh, 



that is, 

s _ Hm rv* . sin {em 

7i = oo|_ 2 6/fl J 

= &L . Hm sin (gZ») . 

2 n = oo B/n 

but by Article 186, (38) 
Km S -^L(^) = 1, 

therefore 



that is, 
5,= lim f— • Sinh ( " /w) l 



, = 00 1 2 



«/» 



_ a?u a 1. sinh (u/n) . 
2 n^oo w/w 

but by Article 186, (38) 

r sinh (Ww) 
hm ^— = !> 

71 = 00 W/W 

therefore 



5,= 



a 2 



(3) *k= 



£rw 



(3) 



196] HYPERBOLIC FUNCTIONS 353 

From (3) we have From (3) we have 

6 = — - ' u = — *> 

a 2 a 2 

and to make the analogy between the circular and hyperbolic func- 
tions complete we may write 

x (2 S\ x , (2 S\ 



y • (2 S 

i = sm'- 



a \a 



2=sinh^V 

a \a 2 J 



x \ a 1 1 x \ a 2 ) 

where S is the area of the cir- where S is the area of the hyper- 
cular sector AOP, and x, y the bolic sector A OP, and x, y the 
coordinates of the point P. coordinates of the point P. 

190. Use of Hyperbolic Functions. A knowledge of hyperbolic 
functions is not only of great importance for the pursuit of higher 
mathematics, but also because of the use which is made of these 
functions in various arts and sciences. Some idea of their usefulness 
may be gathered from the following statement by C. D. Walcott, 
secretary of the Smithsonian Institution. 

"Hyperbolic functions are extremely useful in every branch of 
physics and in the applications of physics, whether to observational 
and experimental sciences or to technology. Thus, whenever an 
entity (such as light, velocity, electricity or radioactivity) is subject 
to gradual extinction or absorption, the decay is represented by 
some form of Hyperbolic Functions. Mercator's projection is like- 
wise computed by Hyperbolic Functions. Whenever mechanical 
strains are regarded as great enough to be measured they are most 
simply expressed in terms of Hyperbolic Functions. Hence geo- 
logical deformations invariably lead to such expressions." 

Because of the great importance of hyperbolic functions, a com- 
plete hyperbolic functions table has recently been published by the 
Smithsonian Institution. 



354 PLANE TRIGONOMETRY [chap, xvi 

Exercise 71 

1. Show that coth -1 ^ = - \ a ^ ■ 

2 6 y — 1 



2. Show that sech -1 3; = log 2_ 



3. Show that csch -1 3; = log —^ ±-2- . 

y 

4. If 5^ denote the area of the hyperbolic sector (Fig. 183) and 

& x —\~ "V 

x, y the coordinates of the point P, show that Sh = — log — — - . 

2 a 

5. Hence find the area of the hyperbolic sector when x — | 
a = 1. ^4?w. 5 A = I log e 3 = 0.5493. 

6. Compute sinh 1 and cosh 1. 

Ans. sinh 1 = 1.1752, cosh 1 = 1.5431. 

7. Compute sinh -1 V and cosh -1 f. 

Ans. sinh -1 -V 2 - = log e 5 = 1.6094, cosh -1 | = log e 3 or log e \ 

= 1.0986 or — 1.0986. 

191. Review. 1. (a) Explain how any equation of the form 
y — f( x ) ma y be graphically represented by a curve, (b) Draw the 
graph for each of the equations x + y = 3, x 2 -\- y 2 = 3 2 , x 2 — y 2 = 3 2 . 

2. (a) Draw on the same sheet of paper (using the same coordi- 
nate axes) the six trigonometric curves, (b) Draw on one sheet of 
paper four curves of the type y = a sin (x + c), giving to a and c the 

values, a — 1, c = o; a = 1, c — - ; a = 3, c = o; # = 3, c = -■ 

2 2 

These curves if properly connected by straight lines form the plan of 
a double- threaded screw of which these curves represent the spirals. 

3. (a) Draw the curve y = log e x. (b) Draw the curve y = log e I - J . 

4. Draw on one sheet the curves y = e kx , putting k successively 
equal to 1, — 1, J and — \. 

5. (a) Construct the curve y = sin x + cos x. 
(b) Draw the curve y = sin x + 2 cos [- 



iqi] HYPERBOLIC FUNCTIONS 355 

6. Draw on one sheet of paper the curves of the six hyperbolic 
functions. 

7. Construct the curve y =e~ x sin x. 

8. (a) If U + iV = (a + ib) + (a + *7>) 2 + (a + *7>) 3 , find Z7 and F. 
(b) Add geometrically 3 + 4 i and 4 — 3 i. 

9. (a) Express 3 + 4 z in the trigonometric form. 

(b) If U + iV = — L r-, find Z7 and V. 

x-\- iy 

(c) If x + £y = r (cos ^ + f sin 0), express z and v in terms of 
r and 6. 

id) In (c) express r and in terms of x and 3;. 

10. (a) Prove that the modulus of a product of two factors is 
equal to the product of the moduli of the factors, (b) Prove that 
the argument of the product of two factors is equal to the sum of 
the arguments of the factors, (c) Prove that the sum of the con- 
jugates of two complex numbers is equal to the conjugate of the sum. 
of the numbers. 

n. (a) State and prove DeMoivre's theorem for positive integral 
exponents, (b) By DeMoivre's theorem find (1 + i) 10 , (J + J i ^3)^ 

12. (a) Find all the roots of the equation x s — 1 = o. 

(b) Find all the roots of the equation x s + 1 = o. 

(c) If o>i and co 2 are the imaginary cube roots of unity, show 
that coi 2 = o?9, W2 2 = wi, 1 + coi + coo = o. 

13. From the relation (cos 6 + i sin 6) 5 = cos 5 6 + i sin 5 6, find 
cos 5 6 and sin 5 d each in powers of sin 6 and cos 6. 

14. (a) What is meant by an infinite series, by a convergent 
infinite series, by a divergent infinite series, by a non-convergent 
infinite series, by an absolutely convergent series, by a semicon- 
vergent series? (b) Give an example of each. 

15. (a) What is meant by the ratio of convergence or test-ratio 
of a series? 

(b) Prove that if the test ratio ultimately approaches some 
value less than unity the series is absolutely convergent. 



356 PLANE TRIGONOMETRY [chap, xvi 

1 6. (a) Write down the first five terms of each of the following 
series: the exponential series, the sine series, the cosine series, the 
logarithmic series, the binomial series, (b) Show that the first 
three are absolutely convergent for all values of x. (c) Show that 
the last two are absolutely convergent so long as x is less than i. 

17. (a) Show that e is greater than 2.5 and less than 2.75. 

(b) Show that ex = 1 -\- x-\ — - H 1- • • • 

2! 2 ! 

(c) By means of the exponential series compute e~ 1= -= 0.3679. 

e 

8t-» ,r . 1 • sin x 1 • lan x 
. Prove that lim ■ = 1, lim = 1. 

x=0 X x=0 X 

19. Compute log e 2, log 10 2, log 3 2. 

20. (a) Show how to compute the sine and cosine of i'. 

' (b) Given sin i' and cos i', show how to find in succession 
the functions of 2', 3', 4/, etc. 

(c) Show how a complete table of functions may be constructed 
when the functions of the angles from o° to 30 are known. 

21. (a) When is a series with complex terms said to be convergent? 

(b) When is such a series said to be absolutely convergent? 

(c) Show that the series 2 u n = 2 [r (cos d + i sin 6)] n is abso- 
lutely convergent. 

22. (a) State and prove Euler's theorem. 

(b) Show that cos 6 = , sin 6 = 



2 2t 

(c) Show that e in+2nii: — — 1, and hence log e — 1 = far + 2 mV. 

23. (a) Define each of the hyperbolic functions. 

(b) Given e = 2.71828, - = 0.36788, compute sinh 2 = 3.6269, 
cosh 2 = 3.7622, sinh \ = 0.5211, cosh \ = 1.1276. 

24. (a) Show that 

cosh u = cos iu, sinh u =— i sin m, tanh u = — i tan m. 

(6) Prove that 

cosh 2 w — sinh 2 u = 1, cosh 2 w + sinh 2 w = cosh 2 w. 



192] HYPERBOLIC FUNCTIONS 357 

25. (a) If y = tanh x, show that x = tanh -1 y = J log T 2 . 

1 - y 

(b) Show that sinh -1 y = log (3; + V ^ 2 +1). 

26. (a) Define the hyperbolic functions geometrically. (6) Men- 
tion some of the uses of hyperbolic functions. 



SPHERICAL TRIGONOMETRY 



CHAPTER I 
INTRODUCTION 

1. Definition of Spherical Trigonometry. If three points on any 
surface are joined by the shortest lines lying in the surface that it 
is possible to draw between these points a triangle is formed. Every 
such triangle has six parts, three sides and three angles. In general 
the sides are not straight lines but geodesic lines, that is, the shortest 
lines that can be drawn on the surface connecting the points. Thus 
every class of surfaces gives rise to a special trigonometry whose 
object is the investigation of the relations between the parts of the 
triangle and the study of the functions necessary for the determin- 
ation of the unknown parts of a triangle from a sufficient number 
of given parts. 

If the surface under consideration is the plane, the geodesies are 
straight lines and the triangles plane triangles, whose properties and 
those of the functions necessary for their solution have been consid- 
ered in plane trigonometry. If the points lie on the surface of a 
sphere the geodesies are arcs of great circles, the triangles are called 
spherical triangles, and the corresponding trigonometry, spherical 
trigonometry. Briefly stated, 

Spherical Trigonometry deals with the relations among the six parts 
of a spherical triangle and the problems which may be solved by means 
of these relations. The most important of these problems consist in 
the computation of the unknown parts of a spherical triangle from 
three given parts. It will be found that the solution of spherical 
triangles requires no functions other than those employed in plane 
trigonometry. 

2. The Uses of Spherical Trigonometry. It is obvious that the 
triangle formed by three points on the earth's surface is not a plane 
triangle but a spherical triangle, for the distances between them are 
measured not along straight lines but along arcs of great circles. It 



2 SPHERICAL TRIGONOMETRY [chap, i 

is only when the distances are comparatively small that the sides may 
be considered straight lines and that the formulas of plane trigonom- 
etry give fairly approximate results. Hence geodetic surveying, 
that is surveying on a large scale, requires a knowledge of spherical 
trigonometry. The same is true of navigation when the bearings and 
distances of distant ports are under consideration. Strictly speaking 
since the earth is not a perfect sphere but a spheroid, such problems 
require a knowledge of spheroidal trigonometry, a branch of trigonom- 
etry whose study demands the introduction of functions other than 
those considered in plane trigonometry, but for many purposes the laws 
of spherical trigonometry give sufficiently accurate approximations. 

While a knowledge of spherical trigonometry is of great importance 
to the surveyor and navigator, it is of even greater importance to the 
astronomer. The positions of all heavenly bodies are referred to the 
surface of an imaginary sphere, the celestial sphere, which encloses 
them all. In fact it is the dependance of astronomy upon spherical 
trigonometry that first led to its study by the ancients, long before 
plane trigonometry was thought of as a separate branch of science. 
Spherical trigonometry is, as it were, the elder sister of plane trig- 
onometry. 

Besides the uses already mentioned, spherical trigonometry fur- 
nishes the best possible review and constitutes one of the most inter- 
esting applications of the principles of plane trigonometry. Spherical 
trigonometry embodies the results of plane trigonometry in much the 
same measure that solid geometry embodies the results of plane 
geometry. 

Finally, spherical trigonometry is worthy of study for its own sake 
because of the marvellous relations which it reveals and the sim- 
plicity, elegance, and beauty of the formulas in which its results are 
embodied. 

3. Spherical Trigonometry Dependent on Solid Geometry. 
Just as plane trigonometry presupposes a certain knowledge of plane 
geometry so spherical trigonometry requires an acquaintance with 
solid geometry, especially with that portion of it which deals with the 
sphere. The student should, therefore, have a textbook on solid 
geometry ready at hand while pursuing this study in order to familiar- 
ize himself anew with the theorems and definitions which are pre- 
supposed in the discussions which follow. He should also provide 
himself with a small wooden or plaster of paris sphere and construct 



4] 



INTRODUCTION 



his figures on it whenever he has difficulty in visualizing the figures 
called for in his study. 

4. Classification of Spherical Triangles. Like plane triangles, 
spherical triangles are classified in two ways: first, with reference to 
the sides and second, with reference to the angles. 

A spherical triangle is said to be equilateral, isosceles, or scalene, 
according as it has three, two, or no equal sides. Since each side of a 





Fig. i. 



Fig- 



spherical triangle may have any value less than 180 ,* one, two, or 
all three of the sides may be quadrants. If one side is a quadrant, 
the triangle is called quadrantal, if two, biquadrantal, if all three, 
triquadrantal. 

A right spherical triangle is one which has a right angle; an oblique 
spherical triangle is one which has none of its angles a right angle. 





Fig- 3- 



Fig. 4- 



Oblique spherical triangles are obtuse or acute according as they have 
or have not an obtuse angle. Since the sum of the angles of a spherical 

* By the number of degrees in an arc we mean, of course, the number of degrees 
in the angle which the arc subtends at the center of the sphere. The number of 
degrees in an arc being given, the length of the arc is at once found from the relation, 
5 = rd, where r is the radius of the sphere and the radian measure of the angle. 
(See PL Trig., Art. 90.) 



SPHERICAL TRIGONOMETRY 



[chap. 



triangle may have any value between 180 and 540 and no single 
angle can exceed 180 , a spherical triangle may have two or even 
three right angles. If it has two right angles it is called birectangular 
(Fig. 1), if three, trirectangular (Fig. 2). For the same reason a spheri- 
cal triangle may have two or even three obtuse angles (Fig. 3). 

If two points on a sphere are at the extremities of the same diam- 
eter any great circle passing through one of the points will pass also 
through the other. Two such points, therefore, cannot be the 
vertices of a spherical triangle, for the great circles connecting these 
points with any third point will coincide and the resulting figure will 
not be a triangle but a lune (Fig. 4). 

5. Co-lunar Triangles. If the arcs AB,AC (Fig. 5) forming two 
sides of any spherical triangle be produced, they will meet again in 
some point A', forming a lune. The third side 
BC divides this lune into two triangles, the origi- 
nal triangle ABC, and the triangle A'BC. The 
triangle A'BC thus formed is said to be co-lunar 
with the triangle ABC. It is obvious that any 
given triangle has three co-luhar triangles, one 
corresponding to each angle of the triangle. Thus 
the triangle ABC (Fig. 5) has the three co-lunar 
triangles A'BC, AB'C, ABC, where A', B' , C 
are the opposite poles of the vertices A, B, C of the triangle ABC. 

Since the angles of a lune are equal, and the sides of the lune semi- 
circles, it follows that the parts of the co-lunar triangles may be im- 
mediately expressed in terms of the parts of the original triangle. If 
we denote the sides of the triangle ABC by a, b, c, and the angles by 
A, B, C, the corresponding parts of the co-lunar triangles are as follows: 




Fig. 5- 



Triangle. 


Sides. 


Angles. 


ABC 

A'BC 

AB'C 

ABC 


a 
a 
i8o°-a 
i8o°-a 


b 
180 - b 

b 
i8o°-b 


c 
i8o°-c 
180 - c 

c 


A 

A 
180 - A 
180 - A 


B 
180 - B 

B 
iSo°-B 


C 

180 - C 

180 - C 

C 



6. Use of Co-lunar Triangles. Any general formula expressing 
a relation between the parts of a spherical triangle must continue true 
when applied to the co-lunar triangles. We may, therefore, sub- 
stitute in any such formula for any two sides and their opposite 



6] INTRODUCTION 5 

angles their supplements, leaving the third side and angle unchanged. 
This process frequently leads to new relations among the parts of the 
triangle. 
Thus, after it has been shown that for any triangle 

a-b C . A + B c 
cos cos — = sin cos - > 

2 2 2 2 

we obtain, by applying this formula to the co-lunar triangle A'BC, 

a - (i8o° - b) i8o° - C . A + (i8o° - B) i8o° - c 
cos cos = sin cos > 

2 2 2 2 

which reduces to the new formula 

. a + b . C A-B . c 

sin sin — = cos sin - • 

2 2 2 2 

Exercise i 

i. Show that every birectangular spherical triangle is also bi- 
/madrantal, and every trirectangular triangle is also triquadrantal. 

2. Prove the converse of the proposition in Problem i. 

3. The co-lunar triangles of any right spherical triangle are right 
spherical triangles, and the co-lunar triangles of any quadrantal 
triangle are quadrantal. 

4. The co-lunar triangles of an equilateral spherical triangle are 
isosceles. 

5. It will be shown later that for any spherical triangle 

a + b . C A + B c 

cos sin — = cos • cos -• 

22 22 

By applying this formula to the co-lunar triangle A'BC show that 

. a-b C . A - B . c 

sin cos — = sin sin — • 

2 2 22 

6. It will be shown later that for any spherical triangle 
sin- = 4 /:l n ( s ~ fl ) sin ^ ~ b> ) 

where 5 = 



V sin a sin b 

a-\- b + c 



2 
By applying this formula to the co-lunar triangle ABC' show that 

'sin 5 sin (s — c) 



C h 



cos— = . 

sin a sin 



SPHERICAL TRIGONOMETRY 



[chap. I 




7. In Fig. 6, ABC is any right spherical triangle, right-angled at 1. 
With B as a pole construct a great circle cutting CB produced in 2 

and BA produced in 3. With i as a pole 
construct a great circle cutting AB produced 
in 4 and CA produced in 5. The resulting 
figure is a curvilinear pentagon bordered by 
five spherical triangles. Show that each of 
these triangles is right-angled and determine 
all their parts as indicated in the figure. (Re- 
mark. The dashes over the letters indicate 
complements, thus A = 90 — A, a= 90 — a t 
c = oo° — c, etc.) 
7. Polar Triangles. If from the vertices of any spherical triangle 
ABC as poles, great circles are drawn they will divide the surface of 
the sphere into eight associated spherical triangles one of which is 
called the Polar of the triangle ABC, and is determined as follows: 

The great circles whose poles are B and C respectively intersect in v 
two points which lie on opposite sides of the arc BC. Let A' be that 
one of these two points which lies on the same side of BC as A . The 
great circles whose poles are C and A respectively intersect in two 



Fig. 6. 






Fig. 7. 



Fig. 8. 



Fig. 9- 



points which lie on opposite sides of the*arc CA . Let B' be that one 
of the two points which lies on the same side of CA as B. Similarly, 
let C be that one of the intersection points of the great circles whose 
poles are A and B respectively, which lies on the same side of the arc 
AB as the vertex C. The triangle whose vertices are A f , B', C is the 
polar of the triangle ABC. 

Just as in triangle ABC we use A, B, C to denote the angles and 
a, b, c to denote the sides opposite these angles, so A', B', C denote 
the angles and a r , b', c' the sides opposite these angles in the polar 
triangle A'B'C. 



g] INTRODUCTION 7 

It is necessary to recall the two fundamental properties of polar 
triangles : 

I. The relation of a triangle to its polar is mutual, that is, if A'B'C 
is the polar of ABC then ABC is the polar of A'B'C . Since each of 
these triangles is the polar of the other, two such triangles are referred 
to as polar triangles. 

II. In two polar triangles each angle is the supplement of the opposite 
side in the other, and each side the supplement of the opposite angle in 
the other. In symbols, 

A + a' = 180 , A' + a = 180 , 

B + V = 180 , B' + b =f 180 , 

C + c' = 180 , C + c = 180 . 

8. The Six Cases of Spherical Triangles. It will be shown 
presently that the six parts of any spherical triangle are so related that 
when any three are given the remaining three can be found. The 
three given parts may be: 

I. The three sides. 

II. The three angles. 

III. Two sides and the included angle. 

IV. Two angles and the included side. 

V. Two sides and the angle opposite one of them. 
VI. Two angles and the side opposite one of them. 

There are six cases of spherical triangles while there are only three 
cases of plane triangles. This is because Cases IV and VI above 
reduce to the same case for plane triangles, since any two angles of 
a triangle determine the third. Also Case II above is ruled out for 
plane triangles since the three angles of a plane triangle determine 
only the shape but not the magnitude of the triangle. 

9. Solution of Spherical Triangles. There are two distinct 
methods of finding the unknown parts of a spherical triangle from three 
known parts: 

I. The Graphic Method. This consists of actually constructing 
the triangle on a material sphere. The unknown parts may then be 
found by measurement. 

II. The Method of Spherical Trigonometry. The unknown parts are 
obtained by computation by means of formulas expressing the rela- 
tion of the unknown parts to the parts which are given. 



8 SPHERICAL TRIGONOMETRY [chap, i 

The first method is purely geometrical and is subject to all the 
errors of construction and inaccuracies of measurement pointed out 
in PL Trig., Art. 3. It is valuable as a rough check on the second 
method rather than as an independent method of solution. 

The second method gives the unknown parts to a degree of ac- 
curacy limited only by the accuracy of the data and the number of 
places of the tables employed in the computation. This is the method 
employed in Geodesy, in Astronomy, and whenever precision is 
necessary or desirable. The derivation of the formulas employed 
by the second method and their application to the solution of the six 
cases of triangles constitutes an important part of Spherical Trigo- 
nometry. 

10. The Use of the Polar Triangle. By the use of the polar 
triangle the second, fourth, and sixth case of spherical triangles may be 
made to depend on the first, third, and fifth respectively. Consider 
for instance Case II, in which the three angles are given. From 
the relations of Art. 7 the three sides of the polar triangle are 
known, this triangle may, therefore, be solved by Case I, and having 
found the angles of this triangle, the sides of the original triangle are 
given by the relations of Art. 7. Similarly, Case IV may be solved 
by Case III, and Case VI by Case V. 

Again by means of the polar triangle any known relation between 
the parts of a triangle may be made to yield another relation, which 
frequently turns out to be new; for a relation which holds for every 
triangle must remain true when applied to the polar, that is, it must 
hold true if we put for each side the supplement of the opposite 
angle and for each angle the supplement of the opposite side. Thus 
if in the formula 

cos J (a — b) cos \ C = sin \ (A + B) cos \ c 

of Art. 6 we put 

a = iSo°-A', b = iSo°-B / , C = i8o°-c', 
A =iSo°-a',B=i8o°- V, c=iSo°-C, 

we obtain 

(i8o°-A f )- (i8o°-5') i8o°-V 
cos cos = 

2 2 

. (i8o°-a')+(i8o°-&') i8o°-C" 
sin cos > 



II] 



INTRODUCTION 



9 




Fig. 10. 



which on reducing becomes 

cos | (A' - B') sin \ c' = sin \ {a! + V) sin \ C, 
or dropping accents 

cos I (A — B) sin \ c = sin J (a + b) sin § C. 

11. Construction of Spherical Triangles. 

Case I. Given the three sides, a, b, c. 

On a sphere lay off an arc BC equal to a* With B as a pole and 
an arc equal to c draw a small circle and with C 
as a pole and an arc equal to b draw another 
small circle. Either of the intersection points, 
A, A f , of these small circles will be the vertex 
of a triangle whose other vertices are B and C 
and whose sides are the three given parts, a, b, c. 

Case II. Given the three angles, A, B, C. 

By Case I construct the polar triangle whose 
sides are 

a=iSo°-A, b=iSo°-B, c = 180 - C. 

The polar of this triangle will be the required triangle. 
Case III. Given two sides and the included angle, a, b, C. 
On a sphere draw an arc CM of a great circle and on it lay off an 
arc CB equal to a. Through C draw an arc CN 
making an angle C with CM.] On CN lay off 
an arc CA equal to b and join A and B by an 
arc of a great circle. Then ABC will be the re- 
quired /triangle. 

Case IV. Given two angles and the included 
side, A, B, c. 

Fig. IIo By Case III construct the polar triangle whose 

two sides and included angle are: 
a = 180 -A; b = 180 - B, C = 180 - c. 
The polar of this triangle will be the required triangle. 

* To lay off an arc equal to a means to lay off an arc of a great circle containing 
a degrees. This may be readily done by means of a strip of paper or cardboard 
equal in length to a semicircumference of the sphere and dividing it into 180 equal 
divisions. Each division will then represent one degree of angular measure on the 
sphere. 

t This is most easily done as follows: From C as a pole draw the arc of a great 
circle. Let M be its intersection with CM. On this arc lay off MN equal to C. 
Join N and C by an arc of a great circle. Then NCM will be the required angle. 
(Why?) 





IO SPHERICAL TRIGONOMETRY [chap, i 

Case V. Given two sides and the angle opposite one of these sides, 
a, b, A. 

We distinguish two cases according as the angle A is acute or 
obtuse. 

I. A acute. 

On a sphere (Fig. 12) draw two arcs, AM and AN, making an angle 

A with each other and let A and A' be their points of intersection, 

On one of these arcs, as AN, lay off AC equal 

to b. With C as a pole and an arc equal to a 

describe a small circle.* In general this circle 

will intersect the arc A M in two points, B and 

B' ', either of which, if its angular distance 

from A is less than 180 , will form the third 

vertex of a triangle whose other two vertices 

are A and C and which will contain the three 
Fig. 12. 

given parts. 
Let p = CD be the arc through C which is perpendicular to AM. 

(a) If a is less than p, the small circle will not intersect AM and 
no triangle exists having the given parts. The solution is impossible. 

(b) If a = p, there is one solution. The resulting triangle has a 
right angle at D. 

(c) If a is greater than p but less than the shorter of the two sides, 
AC = b, CA' = 180 - b, there will be two solutions, ACB and ACB'. 

(d) If a is greater than the shorter of the two sides b and 180 — b 
but less than the longer, there will be one solution. 

(e) If a is greater than the longer of the two sides b and 180 — b 
there will be no solution. 

II. A obtuse. 

Draw the two arcs AM and AN' (Fig. 12), making the angle A with 
each other. On one of these arcs, as AN f , lay off AC equal to b. 
With C as a pole and an arc equal to a describe a small circle which, 
in general, will intersect the arc AM in two points, B and B' , either of 
which, if its angular distance from A is less than 180 , will form the 
third vertex of a triangle whose other two vertices are A and C . 

Let p' = CD be the arc through C which is perpendicular to AM. 
As p is the shortest arc that can be drawn from C to AM, so p' is the 
longest arc that can be drawn from C to AM. 

* This may be done by means of a pair of compasses. 



12] INTRODUCTION II 

(a) If a is greater than p', the small circle will not intersect AM 
and no triangle exists having the given parts. There is no solution. 

(b) If a = p', there is one solution. The resulting triangle has a 
right angle at D. 

(c) If a is less than p' but greater than the longer of the two sides, 
AC = b, C'A' = 180 - b, there will be two solutions, ACB and 
ACB'. 

id) If a is less than the longer of the two sides, b and 180 — b, 
but greater than the shorter, there will be one solution. 

(e) If a is less than the shorter of the two sides, b and 180 — b, 
there will be no solution. 

Case VI. Given two angles and the side opposite one of them, A, B, c. 

By Case V construct the polar triangle whose parts are a = 180 —A, 
b = 180 — B, A = 180 — a. The polar of this triangle will be 
the required triangle. As in Case V, so here there may be either one 
or two solutions or the solution may be impossible. 

12. The General Spherical Triangle. We have defined a spheri- 
cal triangle as the figure formed by joining three points on a sphere, 
which lie not in the same great circle, and no two of which are opposite 
ends of the same diameter, by the shortest great arcs. From this it 
follows that each side of a spherical triangle is less than a semicir- 
cumference, and its angular measure less than 180 . 

Now the great circle drawn through two points is divided by those 
points into two arcs either of which may be considered the arc between 
the two points. If one of these arcs is less 
than 180 the other will be greater than 180 
for their sum is always 360 . Hence if we 
drop the word shortest from the above defini- 
tion, the resulting definition admits triangles 
whose sides have any value between o° and 
360 . Such triangles are called general spheri- 
cal triangles. Since the arc between each two 
vertices may be chosen in two ways there are F - 

eight general triangles having the same three 

vertices. Fig. 13 shows two of these triangles, the triangle AMBC 
and the triangle AM'BC. 

The study of general spherical triangles forms the object of Higher 
Spherical Trigonometry. Their principal applications are found in 
astronomy where it is frequently necessary to consider triangles 




12 SPHERICAL TRIGONOMETRY [chap, i 

whose sides or angles exceed 180 . We observe that every spherical 
triangle, one or more of whose parts exceed 180 , may be solved by means 
of another whose parts are less than 180 , though this is not the simplest 
way of treating such triangles. In the present text we shall limit our 
discussion to triangles which satisfy the first definition, that is, tri- 
angles each of whose parts is less than 180 . 

Exercise 2 

1. Prove the two theorems of Art. 7. 

2. Prove that the polar of a right spherical triangle is quadrantal, 
and conversely, that the polar of a quadrantal triangle is a right 
triangle. 

3. Prove that the polar of a birectangular spherical triangle is 
biquadrantal, and conversely, that the polar of a biquadrantal tri- 
angle is birectangular. 

4. Prove that a trirectangular triangle is its own polar. 

5. If the sides of a triangle are each less than oo° it lies wholly 
within its polar; if each of its sides is greater than 90 its polar lies 
wholly within it. 

6. In any spherical triangle a -\- b -\- c < 360 . By applying this 
relation to the polar show that in every spherical triangle 

180 < A + B + C < 540 . 

7. In every spherical triangle the sum of two sides is greater than 
the third side, that is a + b > c. By applying this relation to the 
polar show that in every spherical triangle the difference between any 
angle and the sum of the other two is less than 180 , that is, A + B — 
C < 180 . 

8. It will be shown later that in any spherical triangle 

cos a = cos b cos c + sin b sin c cos A . 

By applying this formula to the polar triangle show that also 
cos A = — cos B cos C + sin B sin C cos a. 

9. By applying the formulas of Problem 6, Exercise I, to the polar 
triangle, deduce the two new formulas, 



c 

cos - 

2 



_ /cos (S — A) cos (S — B) . c _ I _ cos 5 cos (S — C) 
~V sin A sin B 2 V sin A sin B 

A + B + C 



where S = 



i 2 ] INTRODUCTION 1 3 

10. Construct the triangle called for in Case IV, Art. 11, without 
employing the polar triangle. 

n. In Case V, Art. 11, write out the conditions under which 
the construction admits (a) one solution, (b) two solutions, (c) no 
solution. 



CHAPTER II 

RIGHT AND QUADRANT AL SPHERICAL TRIANGLES* 

13. Formulas for Right Spherical Triangles. Every right 
triangle has a right angle and five other parts which, beginning with 
a side including the right angle, are denoted in order by a, B, c,A, b. 
We shall show that every three of these five parts are so related that 
when two are given the third may be found. Now the above five 
parts admit of ten different sets of three, namely: 

A, a, c; A, b, c; A, a, b; A, B, b; c, a, b; 

B, b, c; B, a, c; B, b, a; B, A, a; c, A, B; 

hence we shall find ten formulas for the right spherical triangle. 
Let ABC, Fig. 14, be a right spherical triangle, C the right angle. Let 

O be the center of the sphere and — ABC the trihedral angle formed 

by the planes of the great circles whose 
arcs are a, b, c, respectively. It is 
shown in geometry that the face angles 
BOC, CO A, AOB are measured by the 
arcs a, b, c, respectively, and that the 
dihedral angles OA, OB, OC are equal 
p. 3 to the angles A, B, C, respectively. 

From any point P in OB draw PR 

perpendicular to OC, and from R draw RS perpendicular to OA. 

Join P and S. Then SR is perpendicular to PR (why?), and PS is 

perpendicular to OA (why?). Hence 

triangle ORP has a right angle at R, 
triangle OSR has a right angle at S, 
triangle OSP has a right angle at S, 
triangle PRS has a right angle at R, 
and angle PSR equals angle A (why ?).f 

* If the class has some knowledge of analytical geometry and the teacher wishes 
to cover the subject in the least time possible, he may omit the work to Art. 26. 
The fundamental relations for the oblique triangle as there developed may be 
specialized for the right triangle by putting C = 90 . 

f See footnote on page 15. 

14 




13] 



RIGHT AND QUADRANTAL SPHERICAL TRIANGLES 



15 



In triangle PRS 



sip. A = 



or 



Interchanging letters 



RP RP/OP sin ROP 
SP SP/OP sin SOP 

sin A = sin a/ sin c. 
sin B = sin &/sin c. 



or 



Interchanging letters 
tan A 



Interchanging letters 




4 ^SR = SR/OS = tan ROS 
cos A - SF - Sp / QS - tSin pos' 

cos A = tan &/tan c, 
cos B = tan a I tan c. 
^P = RP/OR = tan POi? 
S£ == SP/OP ~ sin PCtf' 
tan ^. = tan a/sin &. 
tan .B = tan 6/sin a. 

RP RP 



or 



(1) 

(2) 



(3) 
(4) 



(s) 

(6), 



sin A = 



SP OR 



OR 
OS 



OS 

SP 



->c 



= tan POR • sec ROS ■ cot P05 
tana 1 



Fig, I4# tan c cos b 

whence, substituting the value of tan a/tan c from (4), we have 

sin A = cos B/cos b. (7) 

Interchanging letters sin B = cos A/cos a, (8) 

Once more, cos c = -~p = -^-5 • pr^ = cos POR • cos ROS, 

or cos c = cos a cos &. (9) 

Finally, substituting in (9) for cos a and cos b their values from (7) 

and (8), we obtain 

cos c = cot A cot J5. 

t Let the student who has undue difficulty in per- 
ceiving these relations construct the trihedral angle 
and the corresponding spherical triangle as follows: ^ 
From a piece of cardboard or tin cut out a circle with ** 
any radius. 

Draw four radii OA, OC, OB, OA', making the 
angles 50 , 70 , 77 18', respectively. Cut the circle 
along the radii OA and OA' , and remove the sector 
AMA' . Cut the remaining sector partly through along 
OC and OB and bend the cardboard along these radii 




i6 



SPHERICAL TRIGONOMETRY 



[chap. II 



14. Plane and Spherical Right Triangle Formulas Compared. 

The student will be assisted in remembering the ten formulas of the 
preceding article if he associates them with the corresponding formu- 
las for the plane right triangle, as shown in the following table: 



Plane Right Triangles. 



sin A 



cos A = 



tan A = 



sin A = cos B 



sin B = - 
c 

cos B — - 
c 

tan B = - 
a 

sin B = cos A 



C 2 = a 2 + b 2 

i = cot A cot B 



Spherical Right Triangles. 



sin A = 
cos A = 



tan A = 



sin A = 



sin a 

sin c 
tan b 

tanc 
tanffc 

sinfr 
cos JS 

cos b 



sin_B = 
cos 2? = 



tanJ5 = 



sinJB = 



sin ft 

sin c 
tan a 
tan c 
tan b 

sin a 
cos A 

cos a 



cos c = cos a cos b 
cos c = cot A cot J5 



15. Generalization of the Right Triangle Formulas. In Fig. 
14 the sides a and b are each less than 90 . It remains to show that 
the formulas in Art. 13 hold for all possible values of a and b. 

I. One side adjacent to the right angle greater than qo° and the other 
less than go°. 

In the right triangle ABC (Fig. 16), let a be greater than 90 and 
b less than 90 . The co-lunar triangle AB'C will have a right angle 





at C and the adjacent sides b and a! = 180 — a, each less than 90 . 
We may, therefore, apply the formulas of Art. 13 to this triangle. 
Thus (1) gives 



sin CAB' = 



sin a' sin (180 — a) 



sin a . sin a 

— — , or sin A = — — > 
sine 



sine' sin (180 — c) sine 

that is (1) remains true for the triangle ABC. 

until OA' meets OA. The figure thus formed will be a right trihedral angle, ABC 
will form a right spherical triangle, and the lines PR, RS and PS' will form the 
triangle PRS of Fig. 14. 



i6] 



RIGHT AND QUADRANTAL SPHERICAL TRIANGLES 



17 



Similarly each of the other nine formulas will be found true for the 
triangle A'BC. 

II. Each of the sides adjacent to the right angle greater than go°. 

In the right triangle ABC (Fig. 17), let a and b be each greater than 
90 . The co-lunar triangle ABC will have a right angle at C and the 
adjacent sides a' = 180 — a and b' = 180 — b, each less than 90 . 
We may, therefore, apply the formulas of Art. 13 to this triangle. 
Thus (1) gives 



sin BAC = 



sin a r sin (180 — a) 



sine 



sine 



sin a . sin a 

, or sin A = 



sin c 



sin c 



that is (1) holds true for triangle ABC, and similarly each of the other 
nine formulas will be found true for this case. 

This proves that the formulas of Art. 13 may be applied to the 
solution of every possible right spherical triangle. 

16. Napier's Rules of Circular Parts.* Lord Napier, the in- 
ventor of logarithms, first succeeded in expressing the ten right 
triangle formulas by two simple rules. Let us put 

90 - A = A, 90 - c = c, 90 - B = B, 
then 

sin A = cos A, cos A = sin A, t&nA = cot A, cot A = tan A, 
sin c = cose, etc., sin B = cosB, etc. 

The ten equations of Art. 13 may then be written as follows, the new 
formulas being numbered as in Art. 13. 



sin a = cos A cos c 
sin b = cos B cos c 
sin B = cos A cos b 
sin A = cosB cos a 
sin c = cos a cos b 



(1) sin A = tan b tan c 

(2) sin B = tan a tan c 

(7) sin b = tan a tan A 

(8) sin a = tan b tan B 

(9) sin c = tan A tan B 



Let us now arrange the five parts a, B, c, A, b 
in their order in a circle as in Fig. 18. Any one 
of these five parts, as a, being chosen as the mid- 
dle part, the two next to it, as b and B, are 
called adjacent parts and the remaining two parts, 

* This and the following article may be omitted by tnose a 
who prefer to memorize the preceding ten formulas as sug- 
gested in Art. 14. 




Fig. 18. 



18 ' SPHERICAL TRIGONOMETRY [chap, n 

as A and c, are called opposite parts. Then each of the five equa- 
tions on the right are contained in 

Rule i. The sine of the middle part is equal to the product of the 
tangents of the adjacent parts, 

and the five on the left are contained in 

Rule 2. The sine of the middle part is equal to the product of the 
cosines of the opposite parts. 

These two rules are known as Napier's Rules of the Circular Parts. 
17. Proof of Napier's Rules of Circular Parts. Napier's rules 
are commonly looked upon as memory rules which happen to include 
the ten right triangle formulas. They have been proclaimed the 
happiest example of artificial memory known to man. Because of 
their supposed artificial character their value as an instrument in 
mathematics has been questioned. We shall now show that Napier's 
rules are not mere mnemotechnic aids but constitute a most remark- 
able theorem which admits of rigorous proof. 

Let ABC i be a right spherical triangle, Ci the right angle. With 
B as a pole draw a great circle cutting C\B produced in C 2 and BA 
produced in C 3 . With A as a pole draw a great circle cutting AB 
produced in C 4 and CiA produced in C 5 . The resulting figure is a 

spherical pentagon ABPRS, bordered by five 
triangles I, II, III, IV, V. 

Since B is the pole of the arc C2C3 the angles 
at Ci and Cz are right angles and since A is the 
pole of arc C4C5 the angles at C4 and C5 are 
right angles. The five triangles are, therefore, 
right triangles. 

FjV ™ Since G and C 2 are right angles, S is the pole 

of C1C2 and consequently SC\ and SC2 are quad- 
rants. For like reasons RC3, RC±, PC 5, PC h BC 2 , BC 3 , Ad, AC$ 
are quadrants. 

With these preliminary observations it is now easy to show that 
the five triangles I, II, III, IV, V have the same circular parts taken 
in the same order, while the position of these parts with respect to the 
right angle is different in each of the triangles. 

Let us compare the two triangles ABCi and PRC2 and denote by 
&2, B 2 , C2, Ai, hi the five parts of II which correspond to a, B, c, A, 
b oil. Comparing angular measures we have 




17] 



RIGHT AND QUADRANTAL SPHERICAL TRIANGLES 



J 9 



a 2 = C 2 R = C 2 C 3 - RC, = (180 - B)- 90 = 90 - B = B, 

B 2 = PRC 2 = iSo°-PRS = 180 - C3C4 = 180 - (C 3 B+AC 4 - AS) 

= 180 — (90 + 90 — c) = c, 
c 2 = PR = C A R + PC$ - C4C5 = 90 + 90 - CJLC h 

= 90 + 90 - (180 - A) = A, 
A 2 = RPC 2 = 180 - BPR = 180 - C±C b = 180 - (&A + AC,) 

= 180 - (6 + 90°)= b, 
b 2 = PC 2 = BC 2 -BP = 90 - (CiP - dB) = 90 - (90 - a) = a; 

hence, (h = B, B 2 = c, C 2 =A, A 2 = b, b 2 = a. 

Now the parts of triangle III may be obtained from those of II, the 
parts of IV from those of III, and the parts of V from those of IV, in 
exactly the same way that the parts of II were obtained from those 
of I. Writing corresponding parts under each other, and remember- 
ing that to obtain the circular parts we must replace the hypotenuse 
and angles of each triangle by their complements, we have the follow- 
ing table: 





Actual Parts. 


Circular Parts. 


Triangle I 

Triangle II 


a, B, c, A, b, 
B, c, A, b, a 
c, A, b, a, B 
A, b, a, B, c 

b, a, B, c, A 


a, B, c, A, b 
B, c, A, b, a 
c, A, b, a, B 
A, b, a, B, c 

b, a, B, c, A 


Triangle III 


Triangle IV 


Triangle V 





The column on the right not only shows that each triangle has the same 
circular parts taken in the same order, but also that the middle part 
c of the first triangle is successively replaced by A, b, a, B in the 
other four. Now it was shown in Art. 13 (10), (9), that for the tri- 
angle ABCi, 

cos c = cot A cot B, or sin c = tan A tan B, (I) 



cos c = cos a cos b, or sin c = cos a cos b, 



(id 



hence formulas (I) and (II) hold when any part other than c is taken 
for the middle part, and thus Napier's rules are shown to be neces- 
sarily true. 






20 SPHERICAL TRIGONOMETRY [chap, h 

Exercise 3 

1. Apply the ten formulas for the right spherical triangle to the 
polar and obtain ten formulas for the quadrantal spherical triangle. 

2. Write out the ten equations for the right spherical triangle by 
means of Napier's rules. 

3. From the relation cos c = cos a cos b show that if a right tri- 
angle has only one right angle, the three sides are either all acute, or 
one is acute and the other two obtuse. 

4. From the relation cos A = cos a sin B show that the side a is 
in the same quadrant as the opposite angle A. 

5. If in a right spherical triangle a = c = 90 , prove that cos b = 
cos B. 

6. Also if a = b, prove that cot B — cos a. 

Prove the following relations for the right triangle ABC: 

7. cosM — sin 2 i? = — sin 2 6 sinM. 

8. sin A sin 2 b = sin c sin 2 B. 

9. sin 2 a + sin 2 Z> — sin 2 c = sin 2 a sin 2 £. 

10. sin A cos c = cos a cos B. 

11. sin b = cos c tan a tan B. 

12. sinM cos 2 & sin 2 c = sin 2 c — sin 2 6. 

18. To Determine the Quadrant of the Unknown Parts in a 
Right Spherical Triangle. When an unknown part is found from 
its cosine, tangent, or cotangent, the sign of the function shows whether 
the part is in the first or second quadrant, that is, whether it is less 
than 90 or greater than 90 . In the cases where the unknown part 
is found from the sine, the following theorems enable us to tell, in every 
case in which the triangle has but one solution, whether the part is 
greater or less than 90 . 

I. At least one side of every right spherical triangle is in the first 
quadrant, the remaining two are either both in the first quadrant or both 
in the second. For, since the cosine of an angle in the second quadrant 
is negative, it is plain that the equation 

cos c = cos a cos b (Art. 13 (10)) 

must have either none or two of the angles a, b, c in the second 
quadrant. 

II. Either of the oblique angles of a right spherical triangle is in the 
same quadrant as its opposite side. For since 

sin A = cos JS/cos b (Art. 13 (7)) 



2o] 



RIGHT AND QUADRANT AL SPHERICAL TRIANGLES 21 




and sin A is always positive, it is plain that cos B and cos b must either 
be both positive or both negative, that is, B and b and similarly A 
and a, must be in the same quadrant. 

19. The Ambiguous Case of Right Spherical Triangles. 
When the given parts of a right triangle are an angle and the side 
opposite, the triangle has two solutions. For, the 
given parts being A and a (Fig. 20), the co-lunar 
triangle A'BC as well as the triangle ABC has the 
given parts. Notice that A'B and A'C are the 
supplements of AB and AC, respectively, and that 
angle A'BC is the supplement of angle ABC. 
Both sets of values are given by the formulas, for, 
A and a being given, c, b, and B are found from 
their sines (Art. 13, Equations (1), (5) and (8)). 

20. Solution of Right Spherical Triangles. Napier's rules, or, 
if it is preferred, the ten formulas in Art. 13, enable us to solve every 
conceivable right spherical triangle, two parts being given. The 

procedure in any given case is as follows: 

I. We consider three parts, two of which are 
the given parts and the third the part to be 
found. If these three parts are adjacent we take 
the middle one for the middle part, if two only 
are adjacent we take the remaining one for the 
middle part and by Napier's rules write down 
the formula relating the three parts. 

Thus if A and c are the given parts (Fig. 21), and 
b is to be found, we take A for the middle part and by Napier's first rule, 
sin A = tan b tan c, that is, cos A = tan b cot c. (1) 

If B is to be found, we take c for the middle part, and again applying 
Napier's first rule we have 

sin c = tan A tan B, that is, cos c = cot A cot B. (2) 

If a is the part required, we take a for the middle part, and applying 
Napier's second rule, we have 

sin a = cos A cos c, that is, sin a = sin A sin c. (3) 

II. Next we solve the equation for that function which contains 
the unknown part. Thus to find b, we have from equation (1) above, 
tan b = cos A tan c; to find B we have from (2) cot B = cos c tan A, 
to find a we use equation (3) as it stands. 




Fig. 21. 



22 



SPHERICAL TRIGONOMETRY 



[chap, n 



III. By means of the equations thus obtained and the use of tables 
we compute the unknown parts, remembering, 

(a) If the unknown part is found from its cosine, tangent, or co- 
tangent, the algebraic sign of the function determines the quadrant 
of the angle. 

(b) If the unknown part is found from its sine, the quadrant of the 
angle is determined by one of the theorems of Art. 18. 

(c) If the given parts are an angle and the side opposite, each 
unknown part has two values which are supplements of each other. 

IV. Check. When the unknown parts have been computed, their 
correctness should be checked by the formula obtained by applying 
Napier's rules to these parts. Thus in the above example, after 
b, B, and a have been computed their values must satisfy the formula 
(a being the middle part) 

sin a = tan B tan b, that is, sin a = cot B tan b. 

A 



Example i. 
Given 
A = 67° 34' 40", 

O / // 

c = 137 24 54". 




Required 

b = 160 40' 56 ", 
B = 150 44' 00 " , 

a = 35 42 57 • 



Solution. 

To find b. 

cos A = cot c tan b, 
or, tan b = cos A tan c. 

log cos A = 9.58141 
log tan c = 9.96334^* 
log tan b = 9-54475^ 

b= i6o 4 o , 56 ,/ . 

To find a. ■ 
sin a = sin A sin c. 
log sin A = 9.96586 
log sin c = 9.83038 
log sin a = 9.79624 

a = 35° 42' 57". 
* n written after a logarithm means that the number of which the logarithm is 
taken (in this case tan c) has the negative sign. 



To find B. 
cos c = cot A cos B, 
or, cot B = cos c tan A . 
log cos c = 9.86704^ 
log tan A = 0.38445 
log cot B = 0.25149^ 

B= i5o°44'oo". 

Check, 
sin a = cot B tan b. 
log cot B = 0.25149W 
log tan b = 9-54475^ 
log sin a = 9.79624 (check), 



20] 



RIGHT AND QUADRANTAL SPHERICAL TRIANGLES 



2 3 



In this case, since tan b and cot B are negative, b and B must be 
taken in the second quadrant, while a is taken in the first quadrant 
since by Art. 18 it must be in the same quadrant as the opposite 
angle A . 

Example 2. 

Given /yf/a \ Required 

B = 25° 36' 30", k [M \ A = 81° 48' 30", 

b = 24 20' 45". ( I * J ) ,4' = 98° 11' 30", 

c = 72 30 45", 

c' = 107 29' 15", 

a = 7o°44 / 45 ,/ , 

a' = 109 15' 15". 



To find c. 

sin b = sin B sin c, 

sm^i = cosi5/cos£. or J sin c = sin Z>/ Sm ^« 

log sin b = 9.61515 
colog sin B = 0.36430 








E 


5' 






Fig. 


23 


Solution. 










To find A. 






cosB = 


sin A cos b, 




> 


sin A = 


cos B/cos b. 




log( 


:os B = 


9-955io 




colog 


cos b = 


0.04045 





log sin A = 9.99555 

,4 = 8i° 48' 3o ,/ . 
A'= q8°ii'3o". 

To find a. 

sin a = cot 5 tan b. 
log cot 5 = 0.31940 
log tan b = 9.65560 
log sin a = 9.97500 

a =70° 44' 45". 

a' = 109° 15' 15". 



log sin c 
c 
c' 


= 9-97945 
= 72° 30' 45". 
= 107° 29' 15" 




Check. 


sin a 


= sin c sin A . 


log sin c 


= 9-97945 


log sin A 


= 9-99555 



log sin a =9.97500 (check). 



In this case there are two solutions. By Art. 18 a and A must be 
in the same quadrant, hence the acute values of both a and A belong 
to one triangle and the obtuse values to another. Again, by Art. 18, 
the three sides a, b, c are either all in the first quadrant, or two are 
in the second quadrant, hence c is in the same triangle as a, and c' is 
in the same triangle as a'. 



24 SPHERICAL TRIGONOMETRY [[chap, n 

Exercise 4 

When no answer is given the results must be checked. For the 
number of significant figures to be retained in the answer see PI. 
Trig., Art. 44. 

Solve the following right spherical triangles when the parts given 
are: 

1. a = 8i° 25', b = 101 15'. 

Ans. A = 8i°35', B = ioi°o8', c = 94 40'. 

2. c= 86° 51', B = i8°o4 / . 

Ans. b = i8°o2', a = 86° 41', A = 88° 58'. 

3. a = 70 28', c = 98 18'. 

Ans. A = 72 15', B = 114 17', & = 115 35'. 

4. c = n8°4o', A = i28°oo'. 

Ans. a = 136 16', 5 = 48 24', 5 = 58 27'. 

5. ii = 8i° 13', B = 65 24'. 

i4*w. a = 8o° 20', b = 65 05', c = 85 56'. 

6. b = 112 49', 2? = ioo° 27'. 

,4»s. a = 26 00', ^4 = 27 53', c = no° 24'; 
a' = 154° oo', A' = 152 07', d = 69 36'. 

7. c = 8i° 10', a = ioo° 47'. 

8. A = 75° 23', 5 = 75° 23'. 

9. a = 72 15', B = 8 3 ° 25'. 

10. b = 148 28', B = 101 04'. 

11. a = 43 40.5', c = 98 29.1'. 

y4^5. A = 44 17.0', 2? = 98 11.4', b = ioi°46.3 / . 

12. a = 28 47.0', b = no° 27.3'. 

Ans. A = 30 23. 1', B = ioo° 10.9', c = 107 50.2'. 

13. b = 74° 21.9', 4 = 38 57.2'. 

Ans. B = 8o° 14.7', a = 37 54.1', c = 77 43.3'. 

14. A = 49 15.0', i? = 52 26.0'. 

Ans. a = 34 33.7', b = 36 24.6', c = 48 29.3'. 

15. c = 50 20.2', A = 101 29.4'. 

Ans. a = 131 oi.j',b = 166 29.5', B = 162 20.1'. 

16. a = 32 10.8', A = 42 24.0'. 

Ans. b = 43 34.8', .S = 6o°43.2 / , £=52° 06.0'; 
y = I3 6° 25. 2',^' = 119 16.8', c' = 127 54.0'. 






2i]] RIGHT AND QUADRANTAL SPHERICAL TRIANGLES 25 

17. c = 95 26.2', b = 12 37.8'. 

18. a = 119 56.1', b = i5i°43.6'. 

19. A = 70 56.9', B = 39 25.6'. 

20. 5 = 112 24.8', B = 94 58.9'. 

21. a = 41° 50' 20", 6= 5o°i8' 11". 

iifu. -4 = 49° 19' 29", 5 = 6i° 01' 33 ", c = 6i° 35' 05". 

22. c = no° 46' 20", 5 = 8o° 10' 30". 

ifM. ft = 67° 06' 53", « = i55° 46 r 43", A = 153 58' 24". 

23. & = 9 6 49'59",.4 = 5o i2'o4". 

^4^5. a = 50 oo' 00", i? = 95 14' 41", c = 94 23' 10". 

24. ,4 =46° 59 r 42", 5 =57° 59' i7"- 

Ans. a = 36 27' 00", 6 = 43 33' 30", c = 54 20' 03". 

25. a = 32 09' 17", c = 44 33 f 17". 

Ans. A = 49 20' 16", 6 = 32 41' 00", 5 = 50 19' 16". 

26. b = 160 00' 00", B = 150 oo' 00". 

Ans. a = 140 55' 09", 4 = 112 50' 17", c = 43° 09' 37"; 
a' = 39° 04' 51", A' = 67° 09' 43", C = 136 50' 23". 

27. ^ = 6o 45 / 45^5 = 57 56 , 5 6". 

28. c = 120 23' 56", 4 = 119 34' 42". 

29. a = 11.6 52' 45", b = 16 06' 06". 

30. 4 = 8i° 58' 36", a = 67 20' 30". 

21. Solution of Quadrantal Triangles. The polar of a quadrantal 
triangle is a right triangle which may be solved by the method of 
Art. 20 and from it the required parts of the original quadrantal 
triangle are obtained by means of the relations in Art. 7. Or we may 
apply the right triangle formulas of Art. 13 to the polar and obtain a 
new set of formulas for the solution of any quadrantal triangle. 
Thus formula (1), Art. 13, viz., sin A = sin a/ sin c, when applied to the 
polar triangle becomes sin (180 — a) = sin (180 — A) /sin (180 — C) 
or sin a = sin A /sin C. Similarly we obtain each of the following 
formulas for the solution of quadrantal triangles, C being the angle 
opposite the quadrant c. 

sin a = sin A/sinC (1) tan b = tanl>/sin^4 (6) 

sin b = sin B/sin C (2) sin a = cos b/cos B (7) 

— cos a = tan B/t&n C (3) sin b = cos a/ 'cos A (8) 

— cos b = tan A /tan C (4) — cos C = cos ^4 cos B (9) 
tan a = tan A /sin 5 (5) — cos C = cot a cot & (10) 



26 



SPHERICAL TRIGONOMETRY 



[chap. II 



B' = 145 40', 



Example. Solve the quadrantal triangle in which 
a = 97 24', A = 103 12', c = 90 . 

Solution. The polar triangle has the parts 

A = 180 - 97 24' = 82 36', a = 180 - 103 12' = 76 48', 
C = 180 — 90 = 90 . 

Solving this right triangle by the method of Art. 20 we find 
34 20 , b = 33 37 , c = 79 02 ; 

6' = 146 23', c' = ioo° 58'. 

The required parts of the quadrantal triangle 
are, therefore, 

b = 180 

V = 180 

B = 180 - 33 37 , = : i46 2 3 ', 
5' = 180 - 146 23' = 33 ° 37', 
C = 180 - 79 02' = ioo° 58', 
C = 180 - ioo° 58' = 79 02'. 

Fig. 24 represents both solutions geometrically. 
22. Special Formulas for Angles near 0°, 90° or 180°. An 

angle near o° or 180 can not be accurately determined from its 
cosine, nor an angle near 90 from its sine (see PL Trig., Art. 21); in 
such cases the formulas of Art. 13 are, therefore, no longer adequate. 
The difficulty may be avoided by employing the following formulas: 




or or 

34 20 = 145 40 , 

o / o / 

145 4o = 34 20, 



A near o° or 180 , 

B near o° or 180 , 

a near o° or 180 , 

b near o° or 180 , 

c near o° or 180 , 

A near 90 , 



tan 2 \ A = sin (c — b)/sm (c + b). 
tan 2 \ B = sin (c — a)/sin (c + a). 



tan 



2 1 



= tan \ (c + b) tan i (c — b). 
tan 2 \ b — tan \ (c -f- a) tan \{c — a). 



tan 2 \c = -cos (A + J5)/cos (A - B) 
tan 2 (45° — J A) = tan \ (c — a) /tan J (c + a) 
= tan J (J5 - b) tan J (5 + 6). 
B near 90 , tan 2 (45 — ^ B) = tan J (c — 6)/tan \ (c + &) 

= tan J (^4 — a) tan J (^4 + a). 
a near 90 , tan 2 (45 — \ a) = sin (B — b)/sin (B + 6).. 
5 near 90 , tan 2 (45 — \ b) = sin (.4 — a) /sin (A -\- a), 
c near 90 , tan 2 (45 — J c) = tan J (.4 — a)/tan J (^4 + a) (12) 

= tan § (B - b)/tan \ (B + 5). (13) 



(1) 
(2) 
(3) 
(4) 
(5) 
(6) 
(7) 
(8) 

(9) 
(10) 

(11) 



22] RIGHT AND QUADRANTAL SPHERICAL TRIANGLES 27 

To deduce (1) we have 

cos A = tan ft/ tan c, (Art. 13 (3)) 

1 — cos A tan c — tan & 



1 + cos A tan c + tan ft 
1 — cos A 1 — ( 1 — 2 sin 2 J A) 



(Comp. and div.) 



i + cos4 -i + (2cos*U-i) = tan2 ^' CPL*&fe,Art.in) 

and 

tan c — tan ft _ sin c cos 6 — cos c sin 6 _ sin (c — ft) . 

tan c + tan b sin c cos ft + cos c sin ft sin (c + ft) 

(PI. Trig., Art. 109) 
hence tan 2 \ A — sin(c — ft) /sin (c + ft). 

Again, to deduce (13) we proceed as follows: 

sin c = sin ft/ sin 5, (Art. 13 (2)) 

1 — sine sin B — sin ft ,_ . .. . 

— ; — ; — — ~ — w~, — = — r> (Comp. and div.) 

i + sinc sm B + sin b 

1 — sin c 1 — 2 sin \ c cos J c (cos J c — sin \ c) 2 



1 + sin c 1 -f- 2 sin f c cos § c (cos § c + sin \ c) 2 

(PL Trig., Art. in) 
(1 — tanjc) 2 2/ , . 

= (i + tan|.) 2 = tan(45 "*«>' 

(PL Trig., Art. no) 

and 

sin B — sin ft _ 2 cos | (5 + ft) sin J (5 — ft) __ tan J (5 — ft) . 
sin B + sin ft " 2 sin | (5 + ft) cos \ (B - ft) ~ tan § (5 + ft) ' 

(PL Trig., Art. 113) 
hence tan 2 (45 - \ c) = tan \ (B - ft)/tan \ (B + ft). 

All the other formulas given above may be deduced in a similar 
manner. 

Exercise 5 

1. Solve the quadrantal triangle given in Art. 21 by using formu- 
las (8), (5), and (1) of that article. 

Solve the following quadrantal triangles: 

2. C = 6fi2', ft= i2 3 °48'. 

Ans. B = i 3 o°oo', A = 52° 56', a = 59° S&- 



28 SPHERICAL TRIGONOMETRY [chap, n 

3. C = 141 02.8', A = 142 05.9'. 

Ans. B = 170 15.0', b = 164 29.3', a = 102 10.5'. 

4. a = 1 74 i2 r 49", b = 94 08' 20". 

Ans. A = 175 57' 10", 5 = 135 42' S°"> C = 135° 34' 07". 

5- « = 9 1 " 3°'. b = 9 2 ° 2 4'- 

6. C =136° 14.7', 4 = 141° 45-5'- 

7. a= 112 56' 56", C= 74 45' 36" 

8. In a right spherical triangle one side is 95 52' 15" and the 
hypotenuse is 95 44' 12". Find the angle opposite the given side. 

Ans. 91 15' 01". 

9. Solve the right spherical triangle in which a = 3 7 40' 12", 
c = 37 40' 20". 

Ans. A = 89 25' 32", 5 = oo° 43 r 32", b = oo° 26' 36". 

10. Solve the right spherical triangle in which a = 34 06' 13", 

^ = 34° 07' 41". 

^W5. 6 = 87 32' 39", B = SS° 37' 21", c = 87 58' 00". 

n. Prove formulas (2), (5) and (10), Art. 22. 

12. Verify formulas (3), (6) and (7), Art. 22. 

23. Oblique Spherical Triangles Solved by the Method of 
Right Triangles. Just as every plane triangle can be solved by 
considering it the sum or difference of two right triangles formed by 
drawing a perpendicular from a vertex of the triangle to the opposite 
side or opposite side produced (PL Trig., Art. 52), so likewise every 





Fig. 25. Fig. 26. 

oblique spherical triangle ABC may be solved by considering the 
triangle as the sum (Fig. 25) or the difference (Fig. 26) of the two 
right triangles ACD and BCD formed by the perpendicular arc of a 
great circle drawn from one of the vertices to the opposite side or 
opposite side produced. 



23\ 



RIGHT AND QUADRANTAL SPHERICAL TRIANGLES 



29 



We shall denote by m and n the segments AD and DB into which 
the perpendicular p = CD divides the opposite side c, and by M and 
N the angles ACD and DCB into which the angle C is divided by the 
same perpendicular. We then have 

c = m + n, C = M + N (Fig. 25) ; c = m — n, C = M —N (Fig. 26). 

The method of solving oblique spherical triangles by dividing them 
into right triangles, while exceedingly simple in principle, is not the 
most convenient method nor the method commonly employed in 
actual computation. Better methods will be developed in the next 
chapter and the student is expected to familiarize himself with the 
methods there presented rather than to depend on the method of the 
present article. 

Case III. Given two sides and the included angle, b, c, A. 





Solution. 1. In triangle ACD find p, M and m. 

2. n = c — m (Fig. 27), or n = m — c (Fig. 28). 

3. In triangle BCD find TV, a and B. 

4. C = M + N (Fig. 27), or C = M - N (Fig. 28). 

5. Check. Repeat the solution drawing the perpen- 
dicular from B to the side AC. 

Case IV. Given two angles and the included 
side, B, C, a. 

Solution. Solve the polar triangle by Case 
III and then compute the unknown parts of the 
original triangle. 

Case V. Given two sides and the angle oppo- 
site one of them, a, b, A . 

Solution. In this case there are two solutions, provided that a is 
intermediate in value between p and both b and 180 — b (Art. 11). 




30 SPHERICAL TRIGONOMETRY [chap, n 

i. In triangle ACD find p, M = ACD, and m = AD. 

2. In triangle BCD find N = BCD, B, and n = DB, 
AB'C = 180 - B. 

3. ACB = M + N, ACB' = M - N,AB = m + n, AB' = m - n. 

4. Check. Assume b, c, A as the given parts and find the other 
parts by Case III. 

Case VI. Given two angles and the side opposite one of them, 
A, B, a. 

Solution. Solve the polar triangle by Case V and from it find the 
unknown parts of the original triangle. As there may be two solu- 
tions in Case V so Case VI may have two solutions. 

Case I. Given the three sides, a, b, c. 





Fig- 30- Fig. 31. 

Solution. In the triangle A CD we have by Napier's rule 
sin b = cos p cos m, or cos p = cos b/cos m. 

Similarly we have in the triangle BCD 

sin a = cos p cos n, or cos p = cos a/cos n. 

Hence 

cos a cos m . . . , cos a — cos b cos m — cos n 

r- = , from which — — = ■ 

cos cos n cos a + cos cos m + cos m 

Now 

cos a— cos b — 2sinJ {a+b) sinj (a— b) , . . t , ,. 

i 7 = w — rr^ ti ;\- = — tan| \a-\-b) tanfuz — b), 

cos a+ cos & 2 cos| («+ *) cosj (a - b) 2 v 1 y 2 v / > 

so that 

tan J (a + &) tan | (a — b) = tan | (w + ») tan J (w — w), 

from which 

tan \{m — n) = tan % (a-\- b) tan J (a — b) cot J c, 



23 ] RIGHT AND QUADRANTAL SPHERICAL TRIANGLES 3 1 

if m-\- n = c (Fig. 30), 

or tan \ (w + n) = tan \{a-\-b) tan i(a — b) cot § c, 

if m — n = c (Fig. 31). 

We have, therefore, the following steps: 

1. Find h(m — n) (Fig. 30), or \ (m + n) (Fig. 31), 
from the relation 

tan \ {m — ri) = tan \ {a + &) tan J (a — ft) cot \ c, 
tan J (w + n) = tan J (a + 6) tan J {a — b) cot J c. 

2. m = I (m + ») + J (m — ») , » = \ (m + w) — \ (m — n). 

3. In triangle ACD find A and If. 

4. In triangle BCD find 5 and N. 

5. C = if + N (Fig. 30), or C = M - iV (Fig. 31). 

6. Check. Repeat the solution drawing the perpendicular from 
B on A C or from A on BC. 

Case II. Given the three angles, A, B, C. 

Solution. Solve the polar triangle by Case I, and from it compute 
the unknown parts of the original triangle. 

Exercise 6 

1. Show how Case IV may be solved by means of right triangles 
without using the polar triangle, and outline the steps of the solution. 

2. Prove Bowditch's Rules for Oblique Spherical Triangles which 
may be stated as follows: If a spherical triangle is divided into two 
right triangles by a perpendicular let fall from one of the vertices to 
the opposite side, and if in the two right triangles the middle parts 
are so chosen that the perpendicular is an adjacent part in each 
triangle, then 

The sines of the middle parts in the two triangles are proportional to 
the tangents of the adjacent parts; 
but if the' perpendicular is an opposite part in each triangle, then 

The sines of the middle parts are proportional to the cosines of the 
opposite parts. 

As in the case of Napier's rules, the parts referred to in these rules 
are the circular parts of the two triangles. By the use of Bow- 
ditch's rules the solution of oblique spherical triangles by means of 
right triangles may be somewhat shortened. 



32 SPHERICAL TRIGONOMETRY [chap. I 

Solve the following triangles by means of right triangles: 

3. Given b = 88° 24', c = 56 48', A = 128 16'; 

find B = 65 13', C = 49°28', a = 120 11'. 

4. Given a = 103 44', b = 65 12', C = 97 34'. 

5. Given a = 148 34.4', b = 142 11.6', ^4 = 153 17.6'; 

find c = 62 08.6', B = 148 06.3', C = 130 21.2', 
C ' = 7°i8. 4 ', B f = 31° 53V, C" = 6°i 7 .6'. 

6. Given /I = no°, J5 = 62 , a = 49 . 

7. Given ^ = 8o° 20.2', B = 73 46.7', C = 54 08.5'; 

find a = 64 47.2', b = 6i° 47.3', c = 48 03.4'. 

8. Given a = 31 11' 07", 6 = 32 19' 18", c = 33 15' 21"; 

find A = 59 29' 42", 5 = 62 49' 42", C = 65° 5o' 4 8". 
g. Given a = 87 45' 24", b = 96 12' 15", c = ioo° 08' 56". 
10. Given A = 87 45' 24", B = 96 12' 15", C = ioo° 08' 56". 



CHAPTER III 



PROPERTIES OF OBLIQUE SPHERICAL TRIANGLES 

We shall now develop a number of formulas involving the parts of 
any spherical triangle, from which, if any three parts of the triangle 
are given, the remaining parts may be derived by computation with- 
out first dividing the triangle into right triangles as was done in the 
last article. Then, in order to facilitate the work of computation, 
we shall transform these formulas so as to adapt them to the use of 
logarithms. The actual application of the formulas to the solution 
of triangles we shall reserve for a separate chapter. 

24. The Law of Sines, (a) First Proof. Let ABC be any 
spherical triangle, p the perpendicular from one of the vertices C of 
the triangle to the opposite side AB (Fig. 32) or AB produced (Fig. 

33)- 





Fig. 32 



Fig. 33- 



By Napier's rules, or the formulas of Art. 13, we have 
from triangle A CD ship = sin b sin A, 

and from triangle BCD sin p = sin a sin B (B acute), 

or sin p = sin a sin (180 — B) 

= sin a sin B (B obtuse) . 

Hence, whether the perpendicular falls within the triangle or without, 

we have 

sin p = sin a sin A = sin a sin B. 

Advancing letters, sin c sin B = sin b sin C, 

sin a sin C = sin c sin A 
33 



34 



SPHERICAL TRIGONOMETRY 



[chap, in 



These equations may also be written in the form 

sin a sin b sin c 
sin A ~ sin B sin C 



(i) 



or in words, The sines of the sides of a spherical triangle are propor- 
tional to the sines of the opposite angles. 

(b) Second Proof. Let ABC (Fig. 34) be a spherical triangle and 
O the center of the sphere on which the triangle lies. Draw the radii 
OA, OB, OC. From C draw CD perpendicular to the plane of AOB 
and through CD draw planes CDE and CDF perpendicular to OA 
and OB respectively. Then each of the triangles, OEC, CDE, CDF, 
OFC, is right-angled, the middle letter being in each case at the right 
angle. Also since CF and DF are perpendicular to OB, angle CFD 
is equal to the angle B, and similarly angle CED is equal to the angle 
A. 

Now CD = CE sin CED = CE sin A, 

and CD = CF sin CFD = CF sin B, 

CE = OC sin COE = OC sin b, 

CF = OC sin COF = OC sin a. 

Therefore, substituting in the first two 
equations for CE and CF their values 
from the last two, we have 

OC sin b sin A = OC sin a sin B, 




Fig. 34. 



from which 

sin a/sin A = sin b/sin B. 

25. The Law of Cosines, (a) First Proof. In Figs. 32 and 33 
let us denote AD and DB by m and n respectively. By applying 
Napier's rules, or the formulas of Art. 13, we find 

from triangle BCD cos a = cos p cos n, 

and from triangle ACD cos b = cos p cos m. 

Now n = c — m(B acute), or n — m — c (B obtuse), 

and since cos (c — m) = cos (m — c), we have in either case on elimi- 
nating cos p and putting for n its value 

cos a = cos b cos (c — w)/cos m 

T cos c cos m + sin c sin m 

= cos b 

cos m 

= cos b cos c + cos & sin c tan w. 



26] 



PROPERTIES OF OBLIQUE SPHERICAL TRIANGLES 



35 



But by Napier's rules tan m = tan b cos A, hence substituting this 
value in the last equation and remembering that cos b tan b = sin b } 
we have 

cos a = cos b cos c + sin b sin c cos A. 
Advancing letters, cos b = cos c cos a + sin c sin a cos -B, ■ (2) 

cos c = cos a cos b + sin a sin 6 cos C 

These formulas embody the Law of Cosines: The cosine of any side of 
a spherical triangle is equal to the product of the cosines of the other two 
sides plus the continued product of the sines of these two sides and the 
cosine of the included angle. 





Fig. 32. 



Fig- 33- 



(b) Second Proof. In Fig. 34 draw EG parallel to DF and DH 
perpendicular to EG, then angle DEB. equals angle AOB or c, and we 
have 

HD HD DE CE 



OC DE CE OC 
HD _ OF OG OF 
' OC OC~ OC 



= sin c cos A sin b, 
OG OE 



= cos a — cos c cos b. 



OC OC OC OC OE OC 
Equating these two values of HD/OC and solving for cos a we find 
cos a = cos b cos c + sin b sin c cos A . 

26. Relation Between Two Angles and Three Sides. 

The second of the equations (2) may be written 

cos c cos a + sin c sin a cos B = cos b, 

and the first multiplied by cos c gives 

cos c cos a = cos b cos 2 c + sin b sin c cos c cos A . 
Subtracting the second of these equations from the first gives 
sin c sin a cos B = cos b (1 — cos 2 c) — sin b sin c cos c cos A 



36 



SPHERICAL TRIGONOMETRY 



[chap, m 



Now i — cos 2 c = sin 2 c, hence we may divide the equation by sin c, 
and obtain 

sin a cos B = cos b sin c — sin b cos c cos A. 
Similarly, sin a cos C = cos c sin b — sin c cos & cos A. 
sin 6 cos € = cos c sin a — sin c cos a cos -B, 
sin 6 cos A = cos a sin c — sin a cos c cos J5, 
sin c cos ul = cos a sin & — sin a cos 6 cos C, 
sin c cos -B = cos b sin a — sin b cos a cos C. 



(3) 




=rX 



Fig- 35- 



27. Third Proof of the Fundamental Formulas. The three 
equations (i) Art. 24, (2) Art. 25, and (3) Art. 26, may be derived 
simultaneously by the method of analytical geometry.* Let ABC be 

any spherical triangle. Take O, the center of 
the sphere, for the origin of a system of rec- 
tangular coordinates, the plane of BOA for 
the xy-plane, OB for the direction of the x- 
axis, and the positive z-axis on the same side 
of the plane BOA as the vertex C. Join O 
and C. From C drop the perpendicular CR 
on BOY, and through CR pass a plane per- 
pendicular to OB cutting OB in S. Then the 
triangles CRS and CSO have right angles at R and S respectively, 
and angle RSC equals angle B (why?). Denoting the coordinates of 
C by x, y, z and the distance OC by r, we have 

OS = OC cos COS, or x = r cos a, 

RS = SC cos RSC = OC sin COS cos RSC, or y = r sin a cos 2$, 

RC = SC sin &SC = OC sin COS sin RSC, or 2 = r sin a sin 5. 

If OA had been taken for the #-axis, the z-axis remaining unchanged, 
A and a will change places with B and b respectively, and the y co- 
ordinates will have opposite signs, so that the new coordinates x' , y' , 
z' of C will be 

%' = r cos b, y' —— r sin b cos A , z r = r sin b sin ^4 . 

But these are the transformed coordinates of a system having the same 
z-axis while the x- and y-axes are each turned through an angle c, 

* The student without some knowledge of analytical geometry must content 
himself with the proofs given in the preceding articles and those suggested in the 
exercises which follow. 



29] 



PROPERTIES OF OBLIQUE SPHERICAL TRIANGLES 



37 



hence the coordinates x, y, z and x' , y' , z f are related by the trans- 
formation formulas, 

z = z' ', x = %' cos c — y sin c, y = x' sin c + y' cos c. 

Substituting in these three formulas the values of x, y, z, %', y f , z' in 
terms of r and the parts of the triangle, we have, after dividing out r, 

sin a sin B = sin b sin A, (i) 

cos a — cos b cos c -{- sin & sin c cos ^4, (2) 

sin a cos B = cos b sin c — sin b cos c cos ^4. (3) 

28. Fundamental Relations for the Polar Triangle. If we 

apply the formulas (1), (2), (3) to the polar triangle, by putting a = 
180 — A' ' , ^4 = 180 — a! , etc. (Art. 7), and then drop the accents, 
we find that (1) remains unchanged, while (2) and (3) give rise to the 
new sets of formulas: 



cos A = — cos B cos C + sin B sin C cos 

COS.B = 

cos C = 



a. 



and 



cos C cos A + sin C sin A cos b, 
cos ^ cos -B + sin A sin .B cos c, 

sin ^1 cos b = cos J5 sin C + sin B cos C cos a, 
sin ^. cos c = cos C sin B + sin C cos B cos a, 
sin B cos c = cos CsinA + sin C cos ^4 cos b, 
sin B cos a = cos ^1 sin C + sin .4 cos C cos &, 
sin C cos a = cos ^4 sin B + sin ^4 cos B cos c, 
sin C cos 6 = cos B sin ^L + sin B cos ^ cos c. 



(4) 



(5) 



29. Arithmetic Solution of Spherical Triangles. The funda- 
mental relations (1), (2), (3) enable us to solve every case of oblique 
spherical triangles. 

Case I. Given the three sides, a, b, c. 

1 . The angle A may be found by the law of cosines. 

2. The angles B and C may then be found by the law of sines. 

Case III. Given two sides and the included angle, a, b, C. 

1. The third side may be found by the law of cosines. 

2. The angles A and B may then be found by the law of sines. 

Case V. Given two sides and the angle opposite one of them, a, b, A. 

1. The angle B may be found by the law of sines. 

2. The third side might be found by the law of cosines but since the 
law of cosines involves both sin c and cos c the formula solved for 



& 



SPHERICAL TRIGONOMETRY 



[chap. Ill 



either sin c or cos c would involve radical expressions. These may be 
avoided by using the formula 

cos a cos b — sin a sin b cos A cos B 

C0S C = • 2A • 2/t ' 

i — sm 2 6 sinM 

which is obtained by eliminating sin c from the formulas (2) and (3) 
of Art. 27. 

3. The angle C may now be found by the law of sines. 

Cases II, IV, VI. These may be referred to Cases I, III, V, re- 
spectively, by making use of the polar triangle, or we may apply 
formulas (1), (4), (5). 

While the fundamental relations (1), (2), (3) make it possible to 
solve each of the six cases of triangles, it is clear that (2) and (3) are 
not adapted to logarithmic computation. Therefore, in order to 
facilitate computation, it is desirable to obtain other formulas which 
enable us to use logarithms. Such formulas will be developed in the 
following articles. 

Exercise 7 

1. If a!, b', c' denote the sides of the polar triangle, show that 

sin a : sin b : sin c = sin a! : sin b' : sin c f . 

2. If m is the arc joining the vertex C of a spherical triangle to the 
middle point of the opposite side, show that 

cos a + cos b = 2 cos m cos \ c. 

3. If the bisector of the angle C meets the opposite side in D, show 

that 

sin a : sin b = sin BD : sin AD. 

4. State in words the laws expressed 
by formulas (4) and (5), Art. 28. 

5. In Fig. 36 let EGF be the triangle 
in which a plane drawn perpendicular to 

a an edge OA intersects the trihedral angle. 
Then 

GF 2 = OF 2 + OG 2 - 2 OF . OG - cos a. 
GF 2 = EF 2 + EG 2 - 2EF-EG- cos A. 




Fig. 36. 



Subtracting and observing that OF 2 - EF 2 = OE 2 , OG 2 - EG 2 = OE" 
we find 

2 OF • OG • cos a = 2 OE 2 + 2 EF • EG • cos A. 



30] PROPERTIES OF OBLIQUE SPHERICAL TRIANGLES 39 

which, on dividing by 2 OF • OG, leads to 

cos a = cos b cos c + sin b sin c cos A, 
This constitutes & fourth proof of the law of cosines. 

6. From the law of cosines 

cos A = (cos a — cos b cos c)/sin b sin c, 
show that 

sin 2 A _ 1 — cos 2 ^4 _ 1 — cos 2 a— cos 2 £>— cos 2 c+ 2 cos a cos & cos c 
sin 2 a sin 2 a sin 2 a sin 2 6 sin 2 c 

The expression on the right is symmetrical in a, b, and c, hence 

sin 2 A sin 2 B sin 2 C . . . . sin A sin B sin C 

. „ = . 9 , = . » , from which = -r- -r = 

sin 2 a sin' 1 a sin^ c sin a sin a sin c 

This constitutes a fourth proof of the law of sines. 

7. Prove the relation 

cot a sin b = cot A sin C + cos C cos J. 

Suggestion. Multiply the third of the equations (2), Art. 25, by 
cos b, substitute in the first equation and divide by sin b sin c. 

8. By interchanging and advancing letters write down five other 
equations like that in Problem 7. 

9. Apply the relations of Problems 7 and 8 to the polar triangle. 
Do the resulting equations express new relations? 

10. Given b = 135 , c = 45 , A = 6o°; find the remaining parts 
to the nearest degree. 

Ans. a = 104 , B = 141 , C = 39 . 

n. Given a = 120 , b = 6o°, A = 135 ; find the remaining parts 
to the nearest minute. 

Ans. B = 45 00', c = 78 28', C = 53 o8 r . 

12. Given a = 135 , £ = 135°, c = 45 ; find A, B, C, to the nearest 
minute. Ans. A = B = 114 28', C = 65 32'. 

30. Functions of Half the Angles in Terms of the Sides. 

From the law of cosines 

. COS a — COS b COS C . , . /to m • ■ a \ 

cos A = :— j— ; =1 — 2 sm 2 i^4, (PI. Trig., Art. in) 

sin b sin c 

therefore 

. _ , . cos a — cos b cos c cos (b — c) — cos 

2 Sin 2 ^ A = I ; — =— : = : — 7—. "' 

sin sin c sin b sin c 






SPHERICAL TRIGONOMETRY 



[chap, hi 



40 

Now 

cos (b— c) — cosa=2 sin J (a-\-b— c) sin \ (a— b-\-c) (PL Trig., Art. 113) 

= 2 sin (s — c) sin (s — b), where s = % (a-{- b -\- c), 
therefore 



2 sin 2 1 ^4 



2 sin (5 — b) sin (5 — c) 



sin b sin c 



or 



1 4 — 



sin %A 



-t- 



sin (s — b) sin (s — c) 



sin b sin c 



Similarly, 



s . n|jB _ ^/sin (, - c) sin (« - «) 



sin c sin a 



sin g O = 



V s 



sin (s — a) sin (s — 6) 



sin a sin & 



(6) 



Corresponding formulas for the cosines of half the angles may be 
obtained by applying the formulas (6) to the co-lunar triangles. Thus 
by applying the first formula to the co-lunar triangle AB'C whose 
parts are (Art. 5) 180 - A, B, 180 - C, 180 - a, b, 180 - c, 



we obtain 
Similarly, 



cos h A = 



/ sin s sic 
V sin& 



sin (s — a) 



sine 



cos§J5 



/sins 

V si 



s sin (s — b) 



sin c sin a 



T „ A /sin s sin (s — c) 

COS I C = 1/ : ~— 

** V sin <* sm b 



(7) 



To find tan \ A we divide sin \ A by cos | ^4 and obtain 

tanfc 



tan hA = 

* sin (s — a) 

tan Zc 

2 sin {s — 6) 

. tan k 

tan I C = -r—, r 

2 sin (s — c) 



where 



tank 



= / sin (f 



s — a) sin (* — b) sin (* — c) 



sins 



(8) 



£ is the arcual radius of the small circle inscribed in the triangle 
ABC f for if (Fig. 37) represents the intersection of the arcs bisect- 



3i] 



PROPERTIES OF OBLIQUE SPHERICAL TRIANGLES 



41 



ing the angles of the triangle, OF, the arc drawn from O perpendicular 

to one of the sides as AB, will be the arcual 

radius of the inscribed circle. It follows, just as 

in the case of plane triangles (PL Trig., Art. 68), 

that AF = s — a, hence denoting OF by k and 

applying Napier's rules to the right triangle AOF, 

we have 

sin (s — a) = cot \ A tan k, 

or tan \ A = tan &/sin (s — a). 




Fig- 37- 



31. Functions of Half the Sides in Terms of the Angles. If 

we apply the formulas (6) and (7), Art. 30, to the polar triangle 
(Art. 7), by putting A = 180 - a', a = 180 - A', B = 180 - b', 
etc., dropping the accents in the final results, we obtain 



sin 






sin £ b = 



sin i c = 



V 



— cos S cos (S — A) 

sin B sin C 

— cos S cos (S — B) 

sin C sin A 

— cos S cos (S — C) 

sin A sin B 



(9) 



cos I a = y 



cos I b = 



V 



'cos (S - B) cos (S - C) 
sin 2J sin C 

'cos (# - C) cos (# - 2) 



sin C sin .4 



cos 



hc = s/- 



_ . /cos (# - A) cos (# - B) 



sin ^ sin J3 

S = l(A + B + C). 

From (9) and (10) we find 

tan I a = tan K cos (# — A), 



tan 1 6 = tan K cos (S — B), 
tan I c = tan u; cos (S — C), 



where 



tan K 



-si 



cos S 



cos (s - a) cos (s - B) cos (S - c) 



(10) 



(11) 



42 SPHERICAL TRIGONOMETRY [chap, in 

K is the arcual radius of the small circle circumscribed about the 
triangle ABC, for if O (Fig. 38) is the center of this circle, OA, OB, 
OC the arcs joining the center to the vertices of 
the triangle, OF the perpendicular arc from to 
one of the sides as BC, then OA = OB = OC = K, 
the triangles AOB, BOC, CO A are isosceles, and 
BF = FC = J a. Furthermore 

A = BAO + OAC = ABO + ACO = (B - OBF) 
+ (C - OCF) = B + C- 2 OBF, 

OBF = i(B + C-A) = S-A, 

S = HA+B + C). 

If now we apply Napier's rules to the right triangle BOF, we find 

cos OBF = cot BO tan BF 

or cos (S — A) = cot K tan J a, 

from which tan \ a = tan K cos (S — A). 

Exercise 8 

1. Prove the formula for sin ^ C (Art. 30) directly by using the 
relation 

cos c — cos a cos b 




cos C = 1 — 2 sin 2 J C = 



sin a sin b 



2. Prove the formula for cos | A (Art. 30) directly by using the 
relation 

„ , . cos a — cos b cos c 

cos A = 2 cos 2 # A — 1 = ^ — ^^ , 

sin 6 sin c 

and following the method used in deriving the formula for sin \ A . 

3. Prove the formula for sin J a (Art. 31) directly by using the 
relation 

cos A + cos B cos C 



cos a = 1 — 2 sin 2 J a = 



sin 5 sin C 



4. Prove the formula for cos | o (Art. 31) directly by using the 

relation 

, cos A + cos B cos C 

cos a = 2 cos 2 %a— 1 = ; — p . ~ 

sin B sin C 



32] PROPERTIES OF OBLIQUE SPHERICAL TRIANGLES 43 

5. Derive the formula for tan \ a (Art. 31) by applying the formula 
for tan \ A (Art. 30) to the polar triangle. 

6. Derive the formula for cos \ A by applying the formula for 
sin \ A to the co-lunar triangle ABC . 

7. Apply the formula for sin J ^4 to the co-lunar triangle A 'BC. 
Does the resulting formula express a new relation? 

8. The escribed circles of the triangle ABC are the small circles 
inscribed in the co-lunar triangles A'BC, AB'C, ABC . By applying 
the formula for tan k (Art. 30) to these triangles, show that the arcual 
radii, k a , fa, k c of the escribed circles are given by the formulas 

7 /sin 5 sin (s — b) sin (s — c) . , . 

tan k a = V/ = — ? r~ = sin s tan k A , 

V sin [s — a) 

tan kb = sin s tan \ B, tan k c = sin s tan \ C. 

9. By applying the formula for tan K (Art. 31) to the co-lunar tri- 
angle A'BC, show that the arcual radius of the circle circumscribing 
this triangle is given by the formula 



. TT A / COS (S — A) 4- 1 / C 

tan K * = V - cos 5 cos (S -B) cos (5 - C) = " tan 2 ° /c0S 5 ' 

hence also tan i£# = — tan J &/cos 5", tan Kc = — tan J c/cos 5. 
10. Show that 

2 tan i£ = cot k a + cot && + cot k c — cot &, 
and 2 cot & = tan K A + tan ii b + tan i£c — tan K. 

32. Delambre's (or Gauss's) Proportions. By PI. Trig., Art. 
106, we have 

sin \ (A + B) = sin \ A cos \ B + cos § ^4 sin § B. 

Substituting for sin J A, cos J 5, cos \ A, sin J I>, their values from 
(6) and (7), Art. 30, • 

. 1 , , . ^ 4 /sin (5 — b) sin (5 — c) sin 5 sin (5 — b) 

sinf (4 + 5) = V ^ . f - . 2 

▼ sin a sin sin 2 c 



+v/ 



' sin s sin (5 — a) sin (5 — c) sin (5 — a) 

sin a sin & sin 2 c 
/sin 5 sin (5 — c) sin (5 — b) + sin (5 — a) 



sin a sin & sin c 



, _ sin (s — b) + sin (s — a) 

= COS 4 C ; 

smc 



44 



SPHERICAL TRIGONOMETRY 



[chap, in 



Also by PL Trig., Art. in, 113, 

sin (s — b) + sin (s — a) = 2 sin \ (s — b + s — a) cos § (s — b— s + a) 

= 2 sin I c cos |(a — b), 



and 
hence 



and 



sin c = 2 sin § c cos \ c, 



sin (s — b) + sin (5 — a) cos \ (a — b) 



sine 



cos 



sin}U + £) = 



cos J (a — &) cos J C 



cosf c 



Similarly, we obtain corresponding formulas for cos \ {A ■+• 5), 
sin J (A — B) and cos % (A — B). The four formulas, of which the 
third and fourth may also be obtained by applying the first and sec- 
ond to either one of the co-lunar triangles A'BC or AB'C, may be 
written 

sin \{A -f- B) cos I c = cos § (a — 6) cos § C, 
cos K-4 + B) cos § c = cos I {a + 6) sin | C, 
sin I (-4. — B) sin | c = sin | (a — 6) cos | C, 
cos |(^4 — B) sin § c = sin \ {a + b) sin J C 



(12) 



These formulas are known as Delambre's or Gauss's proportions or 
equations. 

33. Napier's Proportions. , If of the equations (12) we divide the 
first by the second, then the third by the fourth, then the fourth by 
the second, and finally the third by the first, we obtain the following 
four new formulas which are known as Napier's proportions or anal- 
ogies. 

cos I {a — b) 



tan Ua + B) = 



cos \{a + b) 



cot \ C, 



tan|U--B)= sin j; CT -^ cot|C, 
sin I (a + b) 



tan § (a + b) — 



tan h (a — b) = 



COSjK-4 ~ B ) 
COS|U +B) 

sin JU-.B) 
sinJU+JB) 



tan I c, 



tan i c. 



(13) 



The second of these formulas may also be obtained by applying the 
first to either of the co-lunar triangles A'BC or AB'C, and the third 
and fourth by applying the first and second to the polar triangle. 



34] PROPERTIES OF OBLIQUE SPHERICAL TRIANGLES 45 

If we divide the first of the equations (13) by the second, or the 
third by the fourth, we obtain the law of tangents 
tan I (a + b) __ tan § (A + B) 
tan |0 - b) ~ tan \{A - B) ' 

34. Formulas for the Area of a Spherical Triangle. It is 

shown in Solid Geometry that the area of a spherical triangle is given 
by the formula 

T = -78o^' (I4) 

where R is the radius of the sphere, and E° the Spherical Excess 
expressed in degrees, that is E° = A + B + C — 180 . 

If E is the spherical excess expressed in radians, E = E° • 7r/i8o, and 

(14) becomes T= B 2 E. (15) 

For a unit sphere (R = 1) T = E, (16) 

hence we have 

Theorem I. The area of a spherical triangle on a unit sphere is equal 
to the spherical excess expressed in radians. 

Theorem II. The area of a spherical triangle on any sphere is equal 
to the area of the corresponding triangle on a unit sphere multiplied by 
the square of the radius. 

The problem of finding various expressions for the area of a spheri- 
cal triangle resolves itself, therefore, into the problem of finding 
various expressions for the spherical excess E. 

(a) In terms of the angles, A, B, C. 

E=2S-tt, where S = |U + B+ C). (1*7) 

(b) In terms of the sides, a, b, c. 
We have 

sinj E = sin (5 - |ir) = sin [J (A + B) + § (C - *■)] 

= sinJU +5)sinjC- cos| {A -f- B) cos \C. 

Substituting for sin \ (A + B) and cos \ (A + B) their values from 
(12), we have 

sm i E = sm KcosiC _ _ cQsi (a+b)] 

cos I c 

sin J C cos \ C 



cosf c 



j (2 sin J a sin J ft). 



46 SPHERICAL TRIGONOMETRY [chap, ni 

Finally by putting for sin \ C and cos J C their values from (6) and (7) 
we arrive at 

CagnoWs Formula, 

sin I E = i =— T -, (18) 

* 2 cos g a cos J 6 cos J c 

where n = Vsin s sin (s — a) sin (s — b) sin (« — c) " 
Or we may proceed as follows: J (C — E) = \ r — \ (A + B), 
and therefore, sin J (C — E) = cos J (4 + B). 

This value substituted in the second of the equations (12) gives 

sin J (C — E) : sin J C = cos J (a + Z>) : cos J c. 
From this proportion we have by division and composition 

sin J C — sin J (C — E) __ cos J c — cos \ (a + b) 
sin J C + sin § (C — E) cos J c + cos § (a + b) 

On reducing each member of this equation by means of the relations 
of Art. 113 (PL Trig.), we obtain 

tan \ E cot \ (2 C — E) — tan \ s tan \ (s — c) ; 

In like manner, by substituting cos J (C — E) = sin f (4 + B) in the 
first of the equations (12), we find 

tan \ E tan \ (2 C — E) — tan \ (s — a) tan i {s — b); 

hence on multiplying these two equations and extracting the square- 
root we obtain 

Lhuilier's Formula, 

tan I E = Vtan | s tan ±(s — a) tan | (s — 6) tan |(« — c) . (19) 

(c) In terms of two sides and the included angle, a, b, C. 
sin (S-iir) - cos J (4 + B + C) 



tan h E = 



cos (5 - |tt) " sin J (4 + 5 + C) 

sin 1(4 + 5)sinjC- cosi U + 5) cosJC 



sinj {A + 5) cos| C + cosf (4 + B) sin J C 

Substituting for sin J (4 + 5) and cos | (4 + 5) their values from 
(12), we have 

sin \ C cos \ C [cos J (a — b) — cos \ {a + 5)] 



tan hE = 



cos J (a — 6) cos 2 J C + cos J (a + &) sin 2 J C ' 



35] 



PROPERTIES OF OBLIQUE SPHERICAL TRIANGLES 



47 



which readily reduces to 



tan | a tan | b sin C 



tan \ E = 

i + tan | ^ tan J & cos C 



(20) 



35. Plane and Spherical Triangle Formulas Compared. 

The student will have observed that there is a striking resemblance 
between the formulas relating to plane triangles and certain of the 
formulas of the present chapter. In the table below are arranged 



Plane Triangles. 



I. Law of Sines 
a b c 



sin A sin B sin C 

II. Law of Cosines 

a? = b 2 + c 2 
— 2 be cos A 

III. Double Formulas 

sin \ (A - B) • I c 

= \ {a — b) cos ^ C 
cos § (A — B)'\c 
= § (a + 6) sin I C 

IV. Law 0/ Tangents 

§(<* + &) _ tan I U + £) 
£(<*-&) tan iU-5) 

V. Half -angle Formulas 



sm 



14 = 4/ (5 -0) (s^~c) 

V fo 

_ * j s — a) 

V be 



cos |^4 
tan I A = 

k 



i T -\/y 



s — a 



-i/fr 


- 


a) (5 


- 


b)(s- 


c) 


V 

VI. 


4 


rea 


5 






s ^ — 


a 


5 — 


b 


s — c 





y 



Spherical Triangles. 



I. Law of Sines 
sin a sin b sin c 



sin A sin 5 sin C 

II. Law 0/ Cosines 

cos a = cos 6 cos c 

+ sin & sin c cos ^1 

III. Delambre's Proportions 
sin ^ (J. — B) sin § c 

= sin \ (a — b) cos ^ C 
cos I (/I — B) sin |c 

= sin \ {a -\- b) sin | C 

IV. Law of Tangents 

tan |_(g_+_&) _ tan % (A + B) 

tan § (a — b) ~ tan % (A — B) 

V. Half-angle Formulas 
_ 4 /sin (5 — 6) sin (s 



sin |^4 

cos I ^4 
tan ^ vl = 



c) 



=v^ 



sin & sin c 

sin ^ sin (s — a) 

sin 6 sin c 
tan £ 



sin (s — a) 



tan 



, _ 4 /sin (5— a) sin (s—b) sin C?— c) 
V sin .? 



sm s 
VI. ^4rea 



tan x £ 



\A 



5 A s — a L s—b s — 

= \/ tan - tan tan tan 

22 22 

T = r 2 E 



48 SPHERICAL TRIGONOMETRY [chap, in 

in parallel columns the more important formulas for plane triangles 
and the corresponding formulas for spherical triangles. The form of 
some of the formulas for plane triangles has been slightly changed in 
order to manifest the resemblance in the most striking manner. 

36. Derivation of Formulas for Plane Triangles from Those 
of Spherical Triangles. We will now show that the resemblance 
between the two sets of formulas is not accidental but is due to a 
definite relation between plane and spherical triangles. If the vertices 
of a spherical triangle remain fixed while the radius (r) of the sphere 
on which the triangle is situated is indefinitely increased, the spherical 
triangle will approach as a limit the plane triangle having the same 
vertices. Consequently, for the limit r = 00, the formulas for the 
spherical triangle must reduce to those for the plane triangle. 

Let a f , b f , c' represent the sides of the spherical triangle expressed 
in radians, then a' = a/r, b' = b/r, c' = c/r, where a, b, c repre- 
sent the actual lengths of the sides (PL Trig., Art. 90). Also by PL 
Trig., Art. 176, we have 

. , a a z , a* • a* 

sin a = — + etc., cos a = 1 — 9 -\ — — — etc., 

r $lr 3 2 - r A ] - r 

. a a? 

tan a = -\ r + etc. 

r 3r 

and similar expressions for sin b', sin c f , etc. 

These expansions involve the radius of the sphere. If now we 
substitute these expansions in any formula relating to spherical tri- 
angles and evaluate the resulting expression for r = 00, the resulting 
formula will express the corresponding relation between the sides and 
angles of the plane triangle. We will illustrate the method by some 
examples. 

(a) The Law of Sines. 

a a z 

h etc. 

sin A _ sin a! _ sin a/r r 3 ! r 3 "' 

sin B sin b r sin b/r b b d 

r $\r z 

Multiplying both numerator and denominator of the expression on 
the right by r, and making r infinite, we obtain 

sin A a 



sin B b 



the law of sines for plane triangles. 



36] PROPERTIES OF OBLIQUE SPHERICAL TRIANGLES 49 

(b) The Law of Cosines. 

cos a' = cos b' cos d + sin V sin c' cos A 

a b c . . b . c . 

or cos - = cos - cos — h sin - sin - cos A , 

r r r r r 

hence 

'a 2 V / fc 2 , & 4 , W c 2 , c 2 , 



2!r 2 ' 4!r 4 ' \ 2!r 2 \\r± ' /\ 2!r 2 ' 4!r 4 

+ ?"^T7 3+ )(^~^ + ) cosA 

If we multiply both sides of the equation by — 2r 2 , drop the terms 
which are common to both sides of the equation, and then make r 
infinite, we have 

a 2 = b 2 + c 2 — 2 be cos A , the law of cosines for plane triangles. 

(c) The Law of Tangents. 

a±b a+b (a + b)* 
t&njU + B) = tanHa r + &') == 2r 2 r "*" 3(2 r) 3 + 

tani(A-B) tan} (a' - V) . a-b a-b (a-b) s .' 

tan p — rr- + 

2 r 2 r 3(2 r) 6 

Multiplying both numerator and denominator on the right by 2r and 
making r infinite, we have 

tan \ {A + J5) a + b . . , x , , L . . 

1 , — — . = 7 , the law of tangents for plane triangles. 

tan 2 v-A -D ) a 

(d) Area of a Triangle. As a final example we will deduce Hero's 
formula for the area of a plane triangle from Lhuillier's formula for 
the spherical excess. 

Denote a' + b' + c' by 2s', then s' = s/r, s' — a! = {s — a)/r, 
s' — ft = (s — b)/r, etc., and we have from Lhuillier's formula 

4 3'4 3 

's— j> (5— ft) 3 Y 5— c (s~~g) 



V \2r 3(2r) 3 /\ 2r 3 (2 r) d /\2r 3 (2 ry )\ 2r 3 (2 r) d 
Multiplying through by 4r 2 gives 

r 2 £ H -2- + = 

3'4 2 



v/('+^ + )(-«+fc# + )(-'+^ + )( 



-«+<i=4' 



I2r 



50 SPHERICAL TRIGONOMETRY [chap, m 

Now as r approaches infinity, E approaches o, r 2 E remains equal to 
the area of the triangle, hence in the limit 

r 2 E=T= Vs - a) (s - b) (s - c) 
which is Hero's formula. 

Exercise 9 

1. Derive the second of the formulas (12). 

2. Derive the third and fourth of the formulas (12) by applying 
the first and second to the co-lunar triangle AB'C. 

3. Derive the fourth of the formulas (13) by applying the third to 
the co-lunar triangle AB'C. 

4. Derive the fourth of the formulas (13) by applying the second 
to the polar triangle. 

5. Show that the area of the co-lunar triangle A'BC, AB'C, ABC is 
r 2 (2 A — E), r 2 (2 B — E), r 2 {2 C — E), respectively, where E is the 
spherical excess of the triangle ABC. 

6. Prove that 

sin(s — a) + sin(s — b) + sin(s — c) — sin s = 4 sin \a sin \b sin \c. 

7. If S, S A , Sb, Sc denote half the sums of the angles of a triangle 
and its three co-lunars respectively, prove that 

S + $a + Sb + S c = 3 7T. 

8. If E, E A , Eb, Eq denote the spherical excesses of a triangle and 
its three co-lunars respectively, show that E -f- E A + Eb + Eq = 2 t, 
and hence that the sum of the area of these triangles is equal to half 
the area of the sphere. 

9. Deduce the double formula for plane triangles from Delambre's 
formulas for spherical triangles. 

10. Deduce the half-angle formulas for plane triangles from the 
corresponding formulas for spherical triangles. 

n. From the formula cose = cos a cos b for right spherical tri- 
angles deduce the formula c 2 = a 2 + b 2 for plane right triangles. 

12. If K, K A , Kb, Kc denote the arcual radii of a triangle and its 
three co-lunars, show that tan K cot K A cot Kb cot Kc = cos 2 5. 



CHAPTER IV 
SOLUTION OF OBLIQUE SPHERICAL TRIANGLES 

37. Preliminary Observations. In Art. 23 it was shown that 
every spherical triangle may be solved by the method of right triangles. 
Again every spherical triangle may be solved by means of the funda- 
mental relations of Art. 27, as was shown in Art. 29. The purpose of 
the present chapter is to present the most approved methods, which, 
though based on apparently more complicated formulas, require, as 
a rule, the least possible amount of computation, and are, therefore, 
commonly employed by computers. 

The computer will do well to observe the following points: 

(a) The arrangement of the work should be orderly and methodi- 
cal. A complete schedule for the tabular work should be made out 
before the tables are used (PI. Trig., Art. 70). 

(b) It will be well to letter the given parts as in the illustrations 
which follow. Thus if the given parts are two sides and the included 
angle, call the larger of the two sides a, the other b, and the angle C. 
This is easier than to rewrite the formulas so as to involve other 
letters. 

(c) Remember that a small angle cannot be accurately found from 
its cosine, nor an angle near 90 from its sine. (PL Trig., Art. 21.) 
Usually there is a choice of formulas which will enable us to avoid 
any inaccuracies arising from this source. 

(d) Remember also that the answer cannot be more accurate than 
the least accurate of the given parts. It is a false show of accuracy 
to compute the answer to the nearest second when one or more of the 
given parts have a lesser accuracy. (PI. Trig., Art. 44, 19.) 

(e) No result can be relied upon unless it has been checked. When 
the answer is given, that may be looked upon as a check, in all other 
cases the computer must provide a check of his own. 

38. Case I. Given the Three Sides, a, h, c. 

Solution. 

1. To find A, B, C. Use the half-angle formulas (8). 

2. Check. Use the law of sines. 

Note. If one angle only is required it is better to use (6) or (7). 

5 1 



52 



SPHERICAL TRIGONOMETRY 



[chap. IV 



Example. 




Cj;— 


Given 


b/ 


\ a \ 


a = 123 34' 45'Y 






*= 75° 56' 33", 


aL 


-• — - — B/ 


c = 105° oo' 18". 






Solution. 




Fig. 39- 


1. To find A, B, and C. 




, . tan k 


tan J 


D tan& 


tan 2 "^ — • / \ > 

sin (5 — a) 


sin (5 — 6) ' 



To find 

4 = 121° 32' 41", 

5 = 82° 52' S3", 
C= 98° 51' 55". 



tan hC — 



tan& 



tan& 



=\/5i 



sin (s — a) sin (s — b) sin (5 — c) 



sms 



5 = 



sin (s—c) 
a-\-b -\- c 



a = i23°34 , 45 ,/ 

h= 75° 56' 33" 

c= 105 oo' 18" 

25 = 304° 31' 36" 



5=152° 15' 48" 
s — a = 28° 41' 03" 

5 



- b = 76° 

- c = 47° 

2 5= 152° 


19' 

IS' 
15' 


IS" 
3°" 
48" (check) 


1 
2 
1 
2 
1 
2 


4 
B 
C 


= 6o° 
= 41° 
= 49° 


46' 20.7" 
26' 26.4" 

25 57-7 


2. Check. 






sin a 



log sin {s — a) = 9.68122 

log sin (s — b) — 9.98751 

log sin {s — c) — 9.86594 

cologsins = 0.33215 

log tan 2 & = 9.86682 

log tan k = 9.93341 

log tan J A = 0.25219 
log tan \ B = 9.94590 
log tan \ C = 0.06747 

,4 = 121° 32V", 
B= 82° 52' 53", 

C= 98° 51' 55" 



sin& 



sin c 



sin ^4 sin B sin C 

log sin a = 9.92071 log sin b = 9.98680 log sin c = 9.98493 
log sin A = 9.93056 log sin B = 9.99664 log sin C = 9.99478 



9.99015 



9.99016 



9.99015 



Exercise 10 

Solve the following oblique triangles: 

1. Given a = 72° 16', b = 8o° 44', c = 41 18'. 

Arts, A = 73° 38', £ = 96° 12', C = 4i°4<>'. 



39] SOLUTION OF OBLIQUE SPHERICAL TRIANGLES 53 

2. Given a = 109 45', b = 73 56', c = 54 32'. 

3. Given a = 105 06.8', b = 93 39.9', c = 50 20.3'. 

Ans. A = 106 38.0', B = 82 04.4', C = 49 49.2'. 

4. Given a = 27 43.8', & = 49 36.8', c = 55 19.7'. 

5. Given a = 120 22' 40" 6 = in° 34' 27", c = 96 28' 35". 

Ans. A = 126 18' 42", 5 = 119 42' 08", C = in° 51' 42". 

6. Given a = 20 45' 23", 6 = 55 56' 56", c = 67 25' 54". 

7. Given a = 131 35' 04", & = 108 30' 14", c = 84 46' 34", 
4 = 132 14' 21". Find J5 and C. 

8. Given a = 35 ° 30' 24", b = 38 57' 12", c = 56 15' 43". 
Findi? = 47 37' 21". 

39. Case II. Given the Three Angles, A, B, C. 

Solution. 

1. To find a, b, c. Use the half-angle formula (11). 

2. Check. Use the law of sines. 

Note. If one side only is required it is better to use (9) or (10). 

Example. 

Given To find 

A= 121 32' 41", a= i23°34' 4 6", 

B = 82 52' 53-, b= 75° 56' 3*", 

C= 98 5 i'55". c= io 5 °oo'i8". 

Solution. 

1. To find a, b, c. 
tan \ a — tan K cos (5 — ^4), tan \ b = tan i£ cos (S — 5), 
tan \c= tan i£ cos (5 — C), 



v 



-cos 5 e ^4-KB + C 

tan K —\ T7, t\ 77; ^ tt; ^ ' o = ■ 



cos (5 — A) cos (S — B) cos {S — C) 

A = 121 32' 41" 

B= 82 52' 53" 5-^= 30 06' 03.5" 

C= 98°5i r 55^ S-£ = 68° 45' 51.5" 

2 S = 303 17' 29" 5 - C = 52 46' 49-5" 
5 = 151 38' 44.5" S = 151° 38' 44.5" (check) 



54 SPHERICAL TRIGONOMETRY [chap, iv 

log (— cos S) = 9.94450 
log cos (S — A) = 9.93709 colog cos (S — A) = 0.06291 

log cos (S — B) — 9.55896 colog cos (S — B) = 0.44104 

log cos (S — C) = 9.78166 colog cos (S — C) = 0.21834 

log tan 2 K = 0.66679 

log tan K = 0.33340 log tan K = 0.33340 

log tan \ a ■ = 0.27049 \ a = 6i° 47' 23" 

log tan i 6 = 9.89236 \ b = 37 58' 16" 

log tan J c = 0.1 1 506 i c — 5 2 ° 3°' °9" 

a = !2 3 ° 34' 46", b = 75° 56' 32", c = 105 00' 18". 



2. Check. 



sin A _ sin B _ sin C 
sin a sin 6 sin c 



log sin yl = 9.93056 log sin B = 9.99664 log sin C = 9.99478 

log sin a = 9.92071 log sin b = 9.98680 log sin c = 9.98493 

0.00985 0.00984 0.00985 

Note. Since the sum of the angles of a spherical triangle is always 
between 180 and 540 , S is necessarily between 90 and 270 , hence> 
cos 5 is always negative and — cos 5 positive. 

Exercise i i 
Solve the following triangles: 

1. Given A = 74° 40', B = 67 30', C = 49° 50'. 

Am. a = 43 36', b = 41° 21', c = 33 07'. 

2. Given A = 125 54', B = 55 35', C = 45° o 5 r . 

3. Given A = 46 59.3', B = 122 32.6', C = 139 00.3'. 

Ans. a = 59 27.4', b = ii7°o6.2 / , c = 123 2o.o'o 

4. Given A = 47° 34-6', B = 74° 54-7', C = 77° 24-5 r - 

5. Given 4 = 59 55' 10", B = 85 36' 50", C = 59° 55' 10". 

Ans. a = 51 if 31", b = 64° 02' 47", c = 51 17' 31". 

6. Given 4 = 109 35' 56", B = in° 23' 06", C = 86° 49' 19". 

7. Given ^ = 15 38' 06", J5 = 16 o6 r 22", C = 159 44' 26 ,r . 

Find^. Ans. b = 52 05' 54". 

8. Given 4 = 50 45' 23", B = 58 01' 10", C = 87 if 00". 

Find C. 



4o] 



SOLUTION OF OBLIQUE SPHERICAL TRIANGLES 



55 



40. Case III. Given Two Sides and the Included Angle, 
a 9 b, C. 

Solution. 

i. To find A and B. First find J (A + B) and J ( A - B) by the 
first two of Napier's proportions (Art. 33), then 

A = i (A + B) + i (A - B), B = i(A+B)-i(A-B). 

2. To find c. Use either one of Delambre's proportions (Art. 32). 

3. Check. Use the law of sines (Art. 24). 



Example. 






_ .. <7 






Given 




C^r" 






To find 


/ // 
a = no 30 24 , 


c B 


-1= 63° 57' 39" 


b = 3 6°47 , 36", 










B= 35°04'°3" 


C = 135 12' 12". 










C = 13 2° 44' 08" 


Solution. 






Fig. 40. 






1. To find 4 and 5. 










tan§ 


U + B) 


cos \ (a 
cos \ (a 




cot \ C. 


tan J 


(A 


-5) 


sin J (a 
^,*v, 1 ^ 


-j) 


cot \ C. 



i (a - b) = 36° 51' 24", *(« + ») = 73° 39' oo", i C = 67° 36' 06". 



log cos J (a — 6) 
colog cos i (a + b) 
log cot J C 
log tan ^ (A + B) = 0.06872 

h{A + B) =49 3o , 5i ,/ 
^ = 63° 57' 39". 



= 9.90316 log sin \ (a — b) = 9.77802 

= 0.55052 colog sin i (a + b) = 0.01793 
= 9.61504 log cot I C = 9.61504 



log tan \ (A — B)= 9.41099 

i(A-B)= 14 26' 48" 
B = 35° 04' 03". 



2. To find c. 



cos £ c = 



3. Check. 



cos J (a — b) 
sin I (A + B) 



cos ^ C. 



sin a 



sin 6 



sine 



log cos i (a— b) = 9.90316 

colog sin J (A+B) = 0.11886 

log cos \C = 9.58098 

log cos \c = 9.60300 



sin A sin B sin C 

log sin a = 9.97157 

log sin A = 9.95352 

0.01805 



56 SPHERICAL TRIGONOMETRY [chap, iv 

log sin b = 9.77738 

\ c — 66° 22' 04" log sin B = 9.75932 

c — 132 44' 08". 0.01806 

log sin c = 9.86598 
log sin C = 9.84794 
0.01804 
We might have found c from the third or fourth of Napier's propor- 
tions but this would have required us to look up one more logarithm. 

Exercise 12 
Solve the following oblique triangles: 

1. Given a = 140 38', b = 130 28', C = 150 34'. 

Ans. A = i6i°47', B = 157 58', c = 85 20'. 

2. Given a = 103 44', b = 64 12', C = 98 33'. 

3. Given a = 156 12. 2', b = 112 48.6', C = 76 32.4'. 

Ans. A = i54°o4.i / , B = 8f 2.7.1', c = 63 48.8'. 

4. Given a = 27 45.5', b = 22 56.7', C = 156 15.9'. 

5. Given a = 88° 12' 20", b = 124 07' 17", C = 50 02' 02". 

Ans. A = 63 15' 10", B = 132° 17' 59", c = 59 04' 25". 

6. Given a = in° 11' 12", 6 = 137 56' 56", C = 23 15' 48". 

7. Given b = 68° 12' 58", c = 8o° 14' 41", A = if 20' 54". 

Ans. B = 52 05' 54", C = 123 07' 37", a = 20 32' ss ". 

8. Given a = 56 56' 56", c = 156 56' 56", 5 = 9 4° 45' 45"- 

41. Case IV. Given Two Angles and the Included Side, 
A, B, c. 

Solution. 

1. To find a and b. First find | (a + #) and J (<z — £) by the last 
two of Napier's proportions (Art. 33), then 

a = I (a + b) + i(a - b), b = \ (a + b) - \ (a - b). 

2. To find c. Use either one of Delambre's proportions (Art. 32). 

3. Check. Use the law of sines. 

Example. 

Given To find 

A= 63*57' 39", a = no° 30' 23", 

B= 35° 04' 03", b= 36° 47' 37", 

<; =132° 44' 08". C= I35°i2 , i5". 



4i] SOLUTION OF OBLIQUE SPHERICAL TRIANGLES 57 

Solution. 

1. To find a and b. 

, , . 7 N cos \ (A — B) . 
t & n?(a + b) = cosi{A+B) t<mic. 

, , . x sin J 04 — 5) - 
tan,(a-6) = sinJ(il + jB) tanJ C . 

i 04 ~ 5) = 14° 26' 48", i (4 + 5) = 49° 3o' 51", § c = 66° 22' 04". 

log cos I {A — B) = 9.98605 log sin \ (A — B) = 9.39702 

colog cos § 04 + B) = 0.18758 colog sin \ (.4 + 5) = 0.11887 

log tan \c = 0.35896 log tan \ c — 0.35896 

log tan \ {a + b) = 0.53259 log tan \ (a — b) = 9.87485 

i(a+b) = 73° 39' 00" \ (a - b) = 36 51' 23" 

a=iio° 3 o'23". 6 = 36° 47' 37". 

2. To find C. 3. Check. 

. _ sin I (A — B) . , sin A sin 5 sin C 

COS f C = — : — T7 TV" sin 2 c - ~ = ~ — T = ~ 

sin 5(0 — 0) sin a sm b sin c 

log sin i (A - B) = 9.39702 log sin A = 9-9535 2 

colog sin § (a — b) = 0.22199 log sin a = 9.97157 



log sin \ c 


= 9.96196 




9.98195 


log cos | C 


= 9-58097 


log sin B 


= 9-7593 2 


2 ^ 


= 67 36' 07.7" 


log sin & 


= 9-77738 


C 


=135° 12' IS" 




9.98194 






log sin C 


= 9-84793 






log sin c 


= 9-86598 



9.98195 

Exercise 13 
Solve the following triangles: 

1. Given A = 67 30', B = 45 50', c = 74 20'. 

Ans. a = 63 15', b = 53 4 6', C = 52 27'. 

2. Given A = 126 45', B = 49 52', c = 8o° 01'. 

3. Given B = 140 43.2', C = ioo° 04.6', a = 6o° 43.6'. 

Ans. b = 145 55.2', c = 119 22.6', A = 8o° 14.8'. 

4. Given C = 139 25.8', A = 13 56.9', b = 29 00.8'. 

5. Given A = 153° 17' 06", 5 = 78 43' 32", c = 86° 15' 15". 

Ans. a = 88° 12' 19", 6 = 78 15' 41", C = 152 43' 5 2 " 

6. Given a = 50 34' 56", B = 124 10' 10", C = 83 25' 25". 



58 SPHERICAL TRIGONOMETRY [chap, rv 

42. Case V. Given Two Sides and the Angle Opposite One 
of Them, a, b, A, 

In this case there may be two solutions (see Art. n). 

sin b sin A 



i. To find B. Use the law of sines, sin B = 



sin a 



Since B is found from its sine it will in general have two values whose 

. . sin b sin A < .,..<. 

sum is 1 80 . sin B = : = 1, according as sin b sin A = sin a. 

sinfl > =*" 

hence B has two values, one value (90 ), or no real value, according as 
sin b sin A = sin a. 

2. To find C. From the second of Napier's proportions 

, „ sin i (a— b) , , . „ N 
tan J C = -r-f-7 — — r( cot \ (A - B). 

2 sin § + b) 2 v ; 

Since C is less than 180 , tan \ C must be positive. Now a-\-b is 
always less than 360 , therefore sin J (a + b) is always positive, 
hence in order that tan \ C may be positive sin \ (a — b) and 
cot J (A — B) must have like signs. Now \ {a — b) and \ (A — B) 
are each numerically less than 90 , hence in order that sin \ (a — b) 
and cot \ (A — B) may have like signs, | (a — b) and \ (A — B) and 
consequently a — b and A — B must have like signs. If both values 
of B satisfy this condition there are two solutions, if only one value 
of B satisfies this condition there is only one solution, if neither 
value of B satisfies this condition there is no solution. 

3. To find c. From the fourth of Napier's proportions 

, sin J (A + B) ^ , / ,. 
tan 4 c = - — T-r-. ^r tan f (a — b). 

2 sin J {A — B) 2 v J 

4. Check. Use the law of sines, or any other formula involving 
B, C, and c, which has not been previously used. 

The foregoing considerations regarding the number of admissible 
solutions may be summed up into the following: 

Rule. 

a. If sin a < sin b sin A, there is no solution. 

b. If sin a = sin b sin A, there is one solution, B = 90°. 

c. If sin a > sin b sin A , each of the two values of B which gives 

A — B and a — b like signs yields a solution. 



42] 



SOLUTION OF OBLIQUE SPHERICAL TRIANGLES 



59 



Example. 


Given 




a = 


= 62° 


15', 


b = 


o 

= 103 


19', 


A = 


■ 53° 


43'- 




Solution. 

1. To find B. 



sin B = 



sin b sin A 



sin a 



Fig. 41. 



log sin b = 9.98816 

log sin A = 9.90639 

colog sin a = 0.05306 

log sin B = 9.94761 

B = 62 2^' or 





To find 


. 


B 


= 62 : 


>-s', 


C 


= i55 C 


43', 


c 


= 153° 


10', 


B r 


= ii 7 c 


35', 


a 


= 59° 


08', 


c' 


= 70 


27'. 



2. To find C. 
sin J (a- 



62 25' or B' = 117 
3. To find c. 



°35'. 



tan^ C = 



b) 



cot %(A-B). 



, sinJG4+#) -, 7N 
tan^c= - — y7~a — ^rtanf(fl- 0). 



° 39', 



sin±(a+b) * K J ' 2 smi(A-B) 

h(a+b)= 82 47', JU+^)= 58° 04', iU+£') = 

*(<*-*) = -20° 32', |U-5) = - 4 °2i , J i(A-B') = - 3 i° 56'. 

Since the signs of a — b and A — B are alike for both values of B 
there are two solutions. 



log sin J {a — b) = 9.54500^ 
colog sin \ (a + b) = 0.00345 
logcot §04 — £) = 1.11880W 
log cot § 04 — .B') = 0.20534^ 
log tan i C = 0.66725 
log tan 1(7= 9-75379 



log sin \ (A + B) = 9.92874 
colog sin I {A — B) = 1.12005^ 
log tan i (a- b) = 9.57351^ 
log sin i(A + B') = 9.99875 
colog sin \ {A — B') = 0.27660^ 
log tan \ c = 0.62230 
log tan I c' = 9.84886 



\C = 77° 5i-5'- 






\c = 76°34.8 r . 


K'= 2 9 °33-9'- 






1 -/ _ ,-0 1 

2 c — 35 I 3o • 


C = 155 43-0'. 






c = i53°°9-6 r . 


C'= 59°°7-8'- 






c ' = 70 27.0'. 


4. Check. 








sinb sin c 


sin c' 






sin B sin C 


sinC" 




log sin b = 9.98816 log sin c = 


9.65466 


log 


sin c r = 9.97421 


log sin B = 9.94761 log sin C= 


9.61411 


log 


sin C'= 9.93366 



0.04055 



0.04055 



0.04055 



60 SPHERICAL TRIGONOMETRY [chap, iv 

Exercise 14 
Solve the following triangles: 

1. a = 56 40', b = 30 50', A = 103 40'. 
,4tw. B = 36 36', C = 52 00', c = 42 39'. 

2. 5 = 44 45', c = 49° 35', B = 58 56'. (Two solutions.) 

3. a = 148 34.4', b = 142 11.6', .4 = 153 17.6'. 
iliw. 5 = 3i°53-7', C = 6° 17.6', c= 7° 18.3'; 

5' = i 4 8 o6.3 , J C = 130 214', c' = 6 2 °o8.8'. 

4. a = 41 25.8', b = 19 57.9', ^4 = 62°o9.5 / . (One solution.) 

5. a = 67 12' 20", b = 48 45' 40", B = 42 20' 30". 

Arts. A = 55° 39' 57", C = 116 34' 18", c =93° 08' 10"; 
il' = 124 20' 03", C = 24° 32' 15", C = 27 37' 20". 

6. a = 38 10' 10", 6 = 24 56' 45", B = 65° 25' 00". (No solution.) 

43. Case VI. Given Two Angles and the Side Opposite One 
of Them, A, B, a. 

As in Case V so here there may be two solutions. (See Art. 1 1.) 
1. To find b. Use the law of sines, 

sin B sin a 



sin b = 



sin A 



2. To find c. From the fourth of Napier's proportions, 

, sin k (A — B) , , . N 

cot ^ = sini^+^) cotHg ~ 6) - 

3. To find C. From the second of Napier's proportions, 

, _ sin \ (a + b) . , . - . 
cot i C = -r- 2 -? — ! — rx tan \ (A - B). 
2 sin i(a— b) 

4. Check. Use the law of sines, or any other formula involving 
b, c, and C, which has not been previously used. 

To determine the number of solutions we have the following rule 
which is based upon a process of reasoning exactly analogous to that 
employed in establishing the corresponding rule in Case V. 

Rule. 

a. If sin A < sin B sin a, there is no solution. 

b. If sin A = sin B sin a, there is one solution, b = 90 . 

c. If sin A > sin B sin a, each of the two values of b, which gives 
to a — b and A — B like signs, yields a solution. 



43] SOLUTION OF OBLIQUE SPHERICAL TRIANGLES 6 1 

Example. 

Given To find 

^ = 45°3o', b= 33 38', 

£ = 37° 22', c= 59 15', 

a = 40 36'. C = 109 37'. 

Solution. 

1. To find b. 

log sin 5 = 9.78312 i(A + B) = 41 26' 

log sin a = 9.81343 \ (A — B) = 4 04' 
colog sin 4 = 0.14676 i (a -\~ b) = 37 06. 8' 

log sin b = 9.7433 1 J (0 - b) = 3 29.2' 

b = 33° 37-5' *(<* + &') = 93° 29.2' 

6' = 146 22.5' \ (a - b') =-52° 48.2' 

A — B and a — V have unlike signs, hence b' does not yield a 
solution. 

2. To find c. 3. To find C. 

log sin J (A — B) = 8.85075 log sin \ (a + b) = 9.78060 

colog sin J (A + -S) = 0.17931 colog sin J (a — b) = 1.2 1588 

log cot \ (a — b) = 1. 21507 log tan \ (A — B) = 8.85185 

log cot \ c = 0.24513 log cot J C = 9.84833 

f<? = 29° 37.6' \C= 54° 48.45 

c = 59 15. 2' C = io9°36.9' 
4. Check. 

sin B _ sinC log sin B = 9.78312 log sin C = 9.97403 
sin b sine log sin b =9.74331 log sin c =9.93421 

0.03381 0.03382 

Exercise 15 
Solve the following triangles: 

1. A — 36 20', B = 46 30', a = 42 12'. 

Ans. b = 55 19', c = 8i° 19', C = 119 19'; 
V = 124 51', c' = 162 38', C" = 164 44 r . 

2. .4 = 6o° 32', B = 25 56', a = 35 18'. (One solution.) 

3. 4 = 73 11.3', B = 6i° 18.2', a = 46°45.5 / . 

4*w. 5 = 41 52.6', c = 41 35.1', C = 6o° 42.8'. 

4. 4 = 103 56.9', B= 79 35.8', a =127° 45.0'. (Two solutions.) 



62 SPHERICAL TRIGONOMETRY [chap, iv 



5. B = 123 40' 20", C = 159 43' 22", c = 159° 5o r 05". 
Ans. b = 55°52 / 30", a = 137 21' 19", ^ = 137 04' 26"; 

&' = 124 07' 30", a' = 65° 39' 44", ^ = ii3° 39' 16". 

6. il = 70 45 r io ,r , 5 = 119 56' 56", b = 79 45' 02". (No 
solution.) 

44. To Find the Area of a Spherical Triangle. 

Example. 

Given a = 124 12' 31", & = 54 18' 16", c = 97 12' 25". Find 
the spherical excess, and hence the area of the triangle, the radius of 
the sphere being 3959 miles. 

Solution. By Art. 34 we have 

/ — ttR 2 E° 

tanj£ = v tan J s tan§ (s— a) tan J (s— b) tan J (s— c), Z 1 = — — 5- • 

180 

J a = 62 06' 15.5" log tan | s = 0.41426 

\ b = 27 09' 08" log tan \ (s — a) = 9.07809 

i c = 48°3Q / 12-5" log tan | (s — b) = 9.95105 

5 =137° 51' 36" log tan J (5 - c) = 9-56871 

is = 68° 55' 48" log tan 2 \E = 9.01211 

1 (s - a ) = 6° 49' 32.5" log tan \E = 9.50605 

i(s-b) =41° 46' 40" i£ = i 7 46'45" 

J (j - c). = 20° 19' 35.5" E° = 71 07' 00" 

(check) 68° 55' 48" 

log R = 3.59759 

log# 2 = 7.19518 

logTr = 0.49715 

log£° = 1.85197 

colog 180 = 7.74473 

log T = 7.28903 

T = 19455 X io 3 square miles. 

45. Applications to Geometry. 

Exercise 16 

Right Spherical Triangles 

1. The hypotenuse of an isosceles right spherical triangle is 6o°. 
Find the length of the equal sides. Ans. 45 . 

2. Find the relations between each two of the three distinct parts of 
an isosceles right spherical triangle. Ans. cos c — cos 2 # = cot 2 A. 



45] SOLUTION OF OBLIQUE SPHERICAL TRIANGLES 63 

3. Show that no isosceles right spherical triangle can have its 
hypotenuse greater than 90 nor its acute angle less than 45 . 

4. Find the altitude and angle of an equilateral spherical triangle 
whose side is 6o°. Ans. Altitude = 54 44', Angle = 70 32'. 

5. If a is the side, A the angle, and p the altitude of an equilateral 

cos a 
spherical triangle, show that sin \ a sin \ A = J, cos p = 5— • 

COS ~2 Cb 

6. The side of a spherical square (a spherical quadrilateral having 
four equal sides and four equal angles) is 73 41', find the angle and 
length of a diagonal. 

Ans. Angle = 118 04.5', Diagonal 106 16'. 

7. The side of a regular spherical polygon (a spherical polygon 
having n equal sides and n equal angles) is a. Find the angle A of the 
polygon, the perpendicular p from the center of the polygon to one 
of the sides, and the distance r from the center to one of the vertices, 
of the polygon. 

. . ., . cos (w/n) . L A , , / / n . sin \ a 

Ans. sm^A = - ^ — , smp = ta.n^ acot{ir/n), smr = 



cos \ a ' sin (ir/n) 

8. Find the perimeter of the polygon (Problem 7) when p = 90 . 

Ans. 2 7r. 

9. Compute the dihedral angles of- a regular tetrahedron. Of a 
regular dodecahedron. Ans. 70 31' 44", 116 33' $4" . 

Suggestion. With a vertex of the polyhedron as a center describe 
a sphere. The points in which the three edges proceeding from the 
vertex intersect the sphere determine an equilateral spherical triangle 
the sides of which are known. 

10. Compute the dihedral angles of a regular octahedron. Of a 
regular icosahedron. Ans. 109 28' 16", 138 11' 23". 

Exercise 17 
Oblique Spherical Triangles 

1. The three face angles of a trihedral angle are BOC = 84 24', 
COA = 72 18', AOB = 6o° 18'. Find the dihedral angles. 

Ans. OA = 93 40', OB = 72 48', OC = 6o° 3 6 / . 

2. Two planes intersect at an angle of 58 40'. From a point of 
their line of intersection two lines are drawn, one in each plane, 



64 SPHERICAL TRIGONOMETRY [chap, iv 

making the angles 42 ° 30' and 64 24' with the line of intersection. 
Find the angle which the lines thus drawn make with each other. 

Ans. 50 33'. 

3. The great pyramid of Gizeh has a square for its base, and the 
angle between two edges at the vertex measures 96 01. 2' . Find the 
angle which each face makes with the horizon. Ans. 51 51'. 

4. A ten-sided pavilion is covered by a pyramidal roof. Two 
consecutive hips of the roof make an angle of 30 . Find the angle 
between two consecutive faces of the roof. Ans. 159 53'. 

5. The opposite faces of an obelisk are inclined at an angle of 16 . 
Find the face angles at the base of the obelisk and the angle between 
two adjacent faces. Ans. 82 04. 6', 91 06.6'. 

6. The ridges of two gable roofs meet at right angles. The slope 
of each roof is 6o°. Find the angle between the planes of the two 
roofs, and the angle the valley makes with each ridge. 

Ans. 104 26. 6', 63 26.1'. 

7. A mason cuts a stone in the shape of a pyramid with a regular 
Tiexagonal base. The edges are inclined at an angle of 30 with the 
base. Find the angle between two adjacent lateral faces, and the 
inclination of the faces to the base. 

Ans. 149 18.6', 39 13.9'. 

8. If a, /3, 7 are the arcs joining any point in a trirectangular 
triangle to the vertices of the triangle, show that 

cos 2 a + cos 2 jS+ cos 2 7 = 1. 

9. An oblique parallelopiped has the three edges OA = 2.59, 
AB = 3.65, OC = 7.21, and the angles AOB = 72 16', BOC = 8o° 44', 
CO A = 41 i8 ; . Find its volume. Ans. 21.30. 

46. Application to Geography and Navigation. 

Exercise 18 

1. Find the shortest distance measured along a great circle between 
New York, lat. 40 42' 44" N., long. 74 00' 24" W., and San Fran- 
cisco, lat. 37° 47' 55" N., long. 122 24' 32" W., the earth being 
considered a perfect sphere, radius 3959 miles. Ans. 2564 miles. 

2. Find the area of a spherical triangle on the earth's surface 
(r = 3959 miles) whose spherical excess is i°. 

Ans. 273,575 square miles. 



46] SOLUTION OF OBLIQUE SPHERICAL TRIANGLES 65 

3. Compare the shortest distances in degrees of San Francisco, lat. 
37° 47' 55" N., long. 122 24' 32" W., and Seattle, lat. 4 7° 35' 54" N., 
long. 122 19/ 59" W., from Tokio, lat. 35 39' 18" N., long. 139 44' 
30" E. 

4. Find the distance in degrees and the bearing of Rio Janeiro, 
lat. 22 55' S., long. 43 09' W., from Cape of Good Hope, lat. 34 22' 
S., long. 1 8° 30' E. 

Ans. Distance 54 29', Bearing S. 84 45' W. 

5. Find the first and final courses from San Francisco, lat. 37 47' 
55" N., long. 122 24' 32" W., to Yokohama, lat. 35 26' 52" N., long. 
139 38' 41" E. Ans. N. 56 51' W., S. 54°i7'W. 

6. A ship sails on an arc of a great circle a distance of 4150 miles 
from lat. 17 N., long. 130 W., the initial course being S. 54 20' W. 
Taking i° = 69 J miles, what is the latitude and longitude of its final 
position. Ans. Lat. 19 41' S., long. 178 21' W. 

7. A vessel sails from Boston, lat. 42 21' N., long. 7i°o3 / W., to 
Cape Town, lat. 33 56' S., long. 18 28' E. Find at what longitude 
the ship crosses the Equator and its course at this point. 

Ans. Long. 17 48' W., course S. 41 19' E. 

8. Find the distance at which a vessel sailing from Seattle to 
Tokio will cross the 180th meridian and its latitude at the time of 
crossing. (See Problem 3.) 

9. Find the latitude and longitude of the place where a ship sailing 
from Cape of Good Hope to Rio Janeiro crosses the meridian at right 
angles. (See Problem 4.) 

Ans. Lat. 34 43' S., long. 9°i5 r E. 

10. Find the longitude and latitude of the place where a ship 
sailing from San Francisco to Yokohama crosses the meridian at right 
angles. (See Problem 5.) 

11. The continent of Asia has nearly the shape of an equilateral 
triangle, each side being approximately 5500 miles. Find the area 
of the triangle (a) regarded as a plane triangle, (b) regarded as a 
spherical triangle, the radius of the earth being assumed 3960 miles. 

Ans. 13,098,500 square miles; 17,228,400 square miles. 



66 SPHERICAL TRIGONOMETRY [chap, rv 

47. Applications from Astronomy. 

Exercise 19 

(For definitions of terms consult any dictionary or textbook on 
astronomy.) 

1. How many seconds does it take for a star whose declination 
is +64 04' to cross the field of a telescope, the diameter of the field 
being 36'? Ans. 329 seconds. 

2. Find the approximate time of sunrise in Seattle, lat. +47 39', 
on Jan. 15, 1913. Suggestion. Look up the sun's declination. 

Ans. >j h 40. 5 m a.m. local apparent time. 

3. Find the length of the longest day at Seattle, lat. +47 39'. 
Suggestion. When the sun is at its summer solstice its declination 

o _/ 

is 23 27 . 

4. The moon's most northerly declination during this Saros 
occurred on March 19, 1913, and was 28 44' 10". Find approxi- 
mately how long it was below the horizon at San Francisco, lat. 37 
48' 24". Ans. S h 56™. 

5. The zenith distance of the sun was observed to be 45 26' the 
afternoon of a day when its declination was +20 32'. If the latitude 
of the place was +37 io r , what was the local apparent time? 

6. The azimuth of the sun was measured and found to be io° 14.2' 
and its zenith distance 25 12.1' at a time when its declination was 
+ 21 39.2', find the latitude of the place. 

Ans. 46 34. i r . 

7. In Problem 6 find the local apparent time. 

Ans. o h 30™ 13 s . 

8. Ati A i5 m i6.i* local apparent time the altitude of the sun was 
found to be 68° 21' 46" at a time when its declination was +22 41' 30". 
Find the latitude of the place. 

9. In Problem 8 find the azimuth of the sun. 

10. The altitude of the sun was measured and found to be 40 18' 
25" at a place whose latitude is 47 39' 06" at 2 h io w 17.8 s local 
apparent time. Find the sun's declination. 

Ans. +6° 25' 53". 

1 1 . The northeastern end of the canal Phison on Mars is in Martian 
latitude o° 03' N. and longitude 335 io' and the southwestern end 



47] SOLUTION OF OBLIQUE SPHERICAL TRIANGLES 67 

is in latitude 40 08 ' S. and longitude 296 58'. Find the length of 
Phison, the diameter of Mars being 4200 miles. 

Ans. 1946.6 miles. 

12. The declination of Algol is +40 37'; find the azimuth of the 
star when setting at Ann Arbor, lat. +42 if. 

13. The declination of Aldebaran is +16 20.1'; find the azimuth 
of the star when setting at Seattle, lat. 47 39. i'. 

Ans. 114 40.8'. 

14. The declination of Procyon is + 5°26'55"; find the azimuth 
of the star when setting at Chicago, lat. 41 ° 50' 01 " . 

15. The declination of 43H Cephei is now (1913) 85 47'. Find 
its azimuth at Washington, D. C, lat. 38 54', ^ h io w after its meri- 
dian passage. 

16. The declination of -Polaris is now (1913) 88° 50' 38". Find 
its azimuth at Seattle, lat. 47 39' 06", 5^ oi m 20 s after its meridian 
passage. Ans. 178 19' 50". 

17. The right ascension and declination of Regulus are 
a =io A o3 m 44.4 s , 8 =+ 12 23' 34". On May 13, 1913, the moon's 
right ascension and declination were a = g h $8 m 37.3 s , 8 = + 15° 32' 
44". Find the angular distance between the moon's center and 
Regulus. Ans. 3 23' 20". 

18. The obliquity of the ecliptic is now (1913) 23 27 / o2 // . Find 
the celestial latitude and longitude of a* star for which a = $ h 15™ 20 s , 
<5=+36°i7 / 56". 

Ans. /3 = + 17° 33 r 19-7", X = 5^° n' 24.5". 

19. What is the greatest altitude of a star on the equator in the 
meridian of Washington, lat. +38 53' 39"? Ans. 51 o6 r 21". 



TABLE I.* 
COMMON LOGARITHMS OF NUMBERS 

Giving Characteristics and Mantissas of Logarithms of Numbers 
from 1 to 100, and mantissas only of numbers from 100 to 10000. 



LOGARITHMS OF NUMBERS. 



N 


Log. 


N 


Log. 


N 


Log. 


N 


Log. 


1 


0.00000 


26 


1.41497 


51 


1.70757 


76 


1.88081 


2 


0.30103 


27 


1.43136 


52 


1.71600 


77 


1.88649 


3 


0.47712 


28 


1.44716 


53 


1.72428 


78 


1.89209 


4 


0.60206 


29 


1.46240 


54 


1.73239 


79 


1.89763 


5 


0.69897 


30 


1.47712 


55 


1.74036 


80 


1.90309 


6 


0.77815 


31 


1.49136 


56 


1.74819 


81 


1.90849 


7 


0.84510 


32 


1.50515 


57 


1.75587 


82 


1.91381 


8 


0.90309 


33 


1.51851 


58 


1.76343 


83 


1.91908 


9 


0.95424 


34 


1.53148 


59 


1.77085 


84 


1.92428 


10 


1.00000 


35 


1.54407 


60 


1.77815 


85 


1.92942 


11 


1.04139 


36 


1.55630 


61 


1.78533 


86 


1.93450 


12 


1.07918 


37 


1.56820 


62 


1.79239 


87 


1.93952 


13 


1.11394 


38 


1.57978 


63 


1.79934 


88 


1.94448 


14 


1.14613 


39 


1.59106 


64 


1.80618 


89 


1.94939 


15 


1.17609 


40 


1.60206 


65 


1.81291 


90 


1.95424 


16 


1.20412 


41 


1.61278 


66 


1.81954 


91 


1.95904 


17 


1.23045 


42 


1.62325 


67 


1.82607 


92 


1.96379 


18 


1.25527 


43 


1.63347 


68 


1.83251 


93 


1.96848 


19 


1.27875 


44 


1.64345 


69 


1.83885 


94 


1.97313 


20 


1.30103 


45 


1.65321 


70 


1.84510 


95 


1.97772 


21 


1.32222 


46 


1.66276 


71 


1.85126 


96 


1.98227 


22 


1.34242 


47 


1.67210 


72 


1.85733 


97 


1.98677 


23 


1.36173 


48 


1.68124 


73 


1.86332 


98 


1.99123 


24 


1.38021 


49 


1.69020 


74 


1.86923 


99 


1.99564 


25 


1.39794 


50 


1.69897 


75 


1.87506 


100 


2.00000 



* Tables I. and II. reprinted by permission from Murray's 
nometric Tables," published by Longmans, Green & Co. 



: Logarithmic and Trigo- 



COMMON LOGARITHMS OF NUMBERS 



N 


O 


1 


2 


3 


4 


5 


6 


7 


8 


9 


D 


100 

101 

102 
103 


00 000 


043 


087 


130 


173 


217 


260 


303 


346 


389 


43 


432 
860 

01284 


475 
903 
326 


518 
945 
368 


561 
988 
410 


604 

*030 

452 


647 

*072 

494 


689 

*115 

536 


732 

*157 

578 


775 

*199 

620 


817 

*242 

662 


43 
42 
42 


104 
105 
106 


703 

02119 

531 


745 
160 

572 


787 
202 
612 


828 
243 
653 


870 
284 
694 


912 

325 

735 


953 
366 
776 


995 
407 
816 


*036 
449 

857 


*078 
490 
898 


42 
41 
41 


107 
108 
109 

110 

111 
112 
113 


938 

03 342 

743 


979 
383 

782 


*019 
423 

822 


*060 
463 

862 


*100 
503 
902 


*141 
543 
941 


*181 
583 
981 


*222 

623 

*021 


*262 

663 
*060 


*302 

703 

*100 


40 
40 
40 


04139 


179 


218 


258 


297 


336 


376 


415 


454 


493 


39 


532 

922 

05 308 


571 
961 
346 


610 
999 
385 


650 

*038 

423 


689 

*077 

461 


727 

*115 

500 


766 

*154 

538 


805 

*192 

576 


844 

*231 

614 


883 

*269 

652 


39 
39 

38 


114 
115 
116 


690 

06 070 

446 


729 
108 

483 


767 
145 
521 


805 
183 
558 


843 
221 
595 


881 
258 
633 


918 
296 
670 


956 
333 

707 


994 
371 
744 


*032 
408 
781 


38 
38 
37 


117 
118 
119 

120 

121 

122 
123 


819 

07188 

555 


856 
225 
591 


893 
262 

628 


930 
298 
664 


967 

335 
700 


*004 
372 
737 


*041 
408 
773 


*078 
445 
809 


*115 

482 
846 


*151 

518 
882 


37 
37 
36 


918 


954 


990 


*027 


*063 


*099 


*135 


*171 


*207 


*243 


36 


08 279 
636 
991 


314 

672 

*026 


350 

707 

*061 


386 

743 

*096 


422 

778 

*132 


458 

814 

*167 


493 

849 
*202 


529 

884 

*237 


565 

920 

*272 


600 
955 

*307 


36 
35 

35 


124 
125 
126 


09 342 
691 

10 037 


377 
726 
072 


412 
760 
106 


447 
795 
140 


482 
830 
175 


517 
864 
209 


552 

899 
243 


587 
934 

278 


621 

968 
312 


656 

*003 

346 


35 

35 
34 


127 
128 
129 


380 

721 

11059 


415 
755 
093 


449 
789 
126 


483 
823 
160 


517 
857 
193 


551 

890 

227 


585 
924 
261 


619 

958 
294 


653 
992 
327 


687 

*025 

361 


34 
34 
34 


N 


O 


1 


2 


3 


4 


5 


6 


7 


8 


9 


D 


PP 44 43 


42 


41 


40 


39 


38 


37 36 


1 
2 
3 


4.4 

8.8 

13.2 


4.3 

8.6 

12.9 


4.2 

8.4 
12.6 


4.1 

8i 

12. c 


1 
i 


4.0 

8.0 

12.0 


3.9 

7.8 

11.7 


3.8 

7.6 

11.4 




3.7 

7.4 

11.1 


3.6 

7.2 
10.8 


4 
5 
6 


17.6 
22.0 
26.4 


17.2 
21.5 

25.8 


16.8 
21.0 
25.2 


16.4 

20. 1 
24. ( 


I 


16.0 
20.0 
24.0 


15.6 
19.5 
23.4 


15.2 
19. C 
22.8 




14.8 

18.5 
22.2 


14.4 
18.0 
21.6 


7 
8 
9 


30.8 
35.2 
39.6 


30.1 
34.4 
38.7 


29.4 
33.6 
37.8 


28/ 
32. 1 
36. < 


r 


28.0 
32.0 
36.0 


27.3 
31.2 
35.1 


26.6 
30.4 
34.2 




25.9 
29.6 
33.3 


25.2 
28.8 
32.4 



COMMON LOGARITHMS OF NUMBERS 



N 


1 


2 


3 


4 


5 


6 


7 


8 


9 


D 


130 

131 
132 
133 


11394 


428 


461 


494 


528 


561 


594 


628 


661 


694 


33 


727 

12 057 

385 


760 

090 
418 


793 
123 
450 


826 
156 

483 


860 
189 
516 


893 
222 
548 


926 
254 

581 


959 
287 
613 


992 
320 
646 


*024 
352 

678 


33 
33 
33 


134 
135 
136 


710 
13 033 

354 


743 

066 
386 


775 
098 
418 


808 
130 
450 


840 
162 
481 


872 
194 
513 


905 
226 
545 


937 
258 
577 


969 
290 
609 


*001 
322 
640 


32 

32 
3^ 


137 
138 
139 

140 

141 
142 
143 


672 

988 

14 301 


704 

*019 

333 


735 

*051 

364 


767 

*082 

395 


799 
*114 

426 


830 

*145 
457 


862 

*176 

489 


893 

*208 

520 


925 

*239 

551 


956 

*270 
582 


32 
31 
31 


613 


644 


675 


706 


737 


768 


799 


829 


860 


891 


31 


922 

15 229 

534 


953 
259 
564 


983 
290 
594 


*014 
320 
625 


*045 
351 
655 


*076 
381 
685 


*106 
412 
715 


*137 
442 
746 


*168 
473 
776 


*198 
503 
806 


31 
31 

30 


144 

145 
146 


836 

16 137 

435 


866 
167 
465 


897 
197 
495 


927 
227 
524 


957 
256 
554 


987 
286 

584 


*017 
316 
613 


*047 
346 
643 


*077 
376 
673 


*107 
406 
702 


30 
30 
30 


147 

148 
149 

150 

151 
152 
153 


732 

17 026 
319 


761 
056 
348 


791 

085 
377 


820 
114 
406 


850 
143 
435 


879 
173 

464 


909 
202 
493 


938 
231 
522 


967 
260 
551 


997 
289 
580 


29 
29 
29 


609 


638 


667 


696 


725 


754 


782 


811 


840 


869 


29 


898 

18184 

469 


926 
213 

498 


955 
241 
526 


984 
270 
554 


*013 

298 
583 


*041 
327 
611 


*070 
355 
639 


*099 
384 
667 


*127 
412 
696 


*156 
441 

724 


29 
29 
28 


154 
155 
156 


752 

19 033 
312 


780 
061 
340 


808 
089 
368 


837 
117 
396 


865 
145 
424 


893 
173 
451 


921 

201 
479 


949 

229 
507 


977 
257 
535 


*005 
285 
562 


28 
28 
28 


157 
158 
159 


590 

866 

20140 


618 
893 
167 


645 
921 
194 


673 

948 
222 


700 
976 
249 


728 

*003 

276 


756 

*0o0 

303 


783 

*058 

330 


811 

*085 
358 


838 

*112 

385 


28 
27 
27 


N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D 


PP 35 34 33 


32 


31 30 


29 28 27 


1 

2 
3 


3.5 
7.0 

10.5 


3.4 

6.8 
10.2 


3.3 

6.6 
9.9 


3 J 
6.4 
9.( 


I 

[ 


3.1 
6.2 
9.3 


3.0 

6.0 
9.0 


2.9 

5.8 
8.7 


2.8 
5.6 
8.4 


2.7 
5.4 
8.1 


4 
5 
6 


14.0 
17.5 
21.0 


13.6 
17.0 
20.4 


13.2 
16.5 
19.8 


12. t 
16. ( 
19 J 


) 
I 


12.4 
15.5 

18.6 


12.0 
15.0 
18.0 


11.6 

14.5 
17.4 


11.2 
14.0 
16.8 


10.8 
13.5 
16.2 


7 
8 
9 


24.5 
28.0 
31.5 


23.8 
27.2 
30.6 


23.1 

26.4 
29.7 


22. < 
25. ( 
28.i 




21.7 
24.8 
27.9 


21.0 
24.0 
27.0 


20.3 
23.2 
26.1 


19.6 

22.4 
25.2 


18.9 
21.6 
24.3 



COMMON LOGARITHMS OF NUMBERS 



N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D 


160 

161 
162 
163 


20 412 


439 


466 


493 


520 


548 


575 


602 


629 


656 


27 


683 

952 

21219 


710 
978 
245 


737 
*005 

272 


763 

*032 

299 


790 

*059 

325 


817 

*085 

352 


844 

*112 

378 


871 

*139 

405 


898 

*165 

431 


925 

*192 

458 


27 
27 
27 


164 
165 
166 


484 

748 
22 011 


511 

775 
037 


537 
801 
063 


564 

827 
089 


590 
854 
115 


617 
880 
141 


643 

906 
167 


669 
932 
194 


696 

958 
220 


722 
985 
246 


26 
26 
26 


167 
168 
169 

170 

171 
172 
173 


272 
531 
789 


298 
557 

814 


324 
583 

840 


350 
608 
866 


376 
634 
891 


401 
600 
917 


427 
686 
943 


453 

712 
968 


479 
737 
994 


505 

763 

*019 


26 
26 
26 


23 045 


070 


096 


121 


147 


172 


198 


223 


249 


274 


25 


300 
553 
805 


325 

578 
830 


350 
603 
855 


376 
629 

880 


401 
654 
905 


426 
679 
930 


452 
704 
955 


477 
729 
980 


502 

754 

*005 


528 

779 

*030 


25 
25 

25 


174 
175 
176 


24 055 
304 
551 


OSO 
329 
576 


105 
353 
601 


130 

378 
625 


155 
403 
650 


180 
428 
674 


204 
452 
699 


229 

477 
724 


254 

502 

748 


279 
527 

773 


25 

25 
25 


177 
178 
179 

180 

181 
182 
183 


797 
25 042 

285 


822 
066 
310 


846 
091 
334 


871 
115 
358 


895 
139 

382 


920 
164 
406 


944 

188 
431 


969 
212 
455 


993 

237 
479 


*018 
261 
503 


25 

24 
24 


527 


551 


575 


600 


624 


648 


672 


696 


720 


744 


24 


768 

26 007 

245 


792 
031 
269 


816 
055 
293 


840 
079 
316 


864 
102 
340 


888 
126 
364 


912 
150 

387 


935 
174 
411 


959 
198 
435 


983 
221 
458 


24 
24 
24 


184 
185 
186 


482 
717 
951 


505 
741 
975 


529 

764 

998 


553 

788 
*021 


576 

811 

*045 


600 

834 
*068 


623 

858 
*091 


647 

881 

*114 


670 

905 
*138 


694 

928 
*161 


24 
23 
23 


187 
188 
189 


27184 
416 
646 


207 
439 
669 


231 

462 
692 


254 

485 
715 


277 
508 
738 


300 
531 
761 


323 
554 

784 


346 

577 
807 


370 
600 
830 


393 
623 

852 


23 
23 
23 


N 


1 


2 


3 


4 


5 


6 


7 


8 


9 


D 


PP 27 


26 


25 




24 


23 22 


1 

2 
3 


2/ 
5.4 

8.] 


r 
[ 


2.6 
5.2 

7.8 




2.5 
5.0 
7.5 






2.4 
4.8 
7.2 




2.3 

4.6 
6.9 


2.2 
4.4 
6.6 


4 
5 
6 


10. t 
13.1 
16. 1 




10.4 
13.0 
15.6 




10.0 
12.5 
15.0 






9.6 
12.0 
14.4 




9.2 
11.5 
13.8 


8.8 
11.0 
13.2 


7 
8 
9 


18.1 
21.( 
24.1 


! 


18.2 
20.8 
23.4 




17.5 

20.0 
22.5 






16.8 
19.2 
21.6 




16.1 

18.4 
20.7 


15.4 
17.6 
19.8 



COMMON LOGARITHMS OF NUMBERS 



N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D 


190 

191 
192 
193 


875 


898 


921 


944 


967 


989 


*012 


*035 


*058 


*081 


23 


28103 
330 
556 


126 

353 
578 


149 
375 

601 


171 
398 
623 


194 
421 
646 


217 
443 
668 


240 
466 
691 


262 
488 
713 


285 
511 

735 


307 
533 

758 


23 
23 
22 


194 
195 
196 


780 

29 003 

226 


803 
026 

248 


825 
048 
270 


847 
070 
292 


870 
092 
314 


892 
115 
336 


914 
137 

358 


937 
159 

380 


959 

181 
403 


981 
203 
425 


22 
22 
22 


197 
198 
199 

200 

201 
202 
203 


447 

667 

885 


469 

688 
907 


491 

710 
929 


513 

732 
951 


535 
754 

973 


557 

776 
994 


579 

798 

*016 


601 

820 
*038 


623 

842 
*060 


645 

863 

*081 


22 
22 
22 


30103 


125 


146 


168 


190 


211 


233 


255 


276 


298 


22 


320 
535 
750 


341 
557 

771 


363 

578 
792 


384 
600 
814 


406 
621 
835 


428 
643 
856 


449 
664 

878 


471 
685 

899 


492 

707 
920 


514 
728 
942 


22 
21 

21 


204 
205 
206 


963 

31175 

387 


984 
197 
408 


*006 
218 
429 


*027 
239 
450 


*048 
260 
471 


*069 
281 
492 


*091 
302 
513 


*112 
323 
534 


*133 
345 
555 


*154 

366 
576 


21 
21 

21 


207 
208 
209 

210 

211 
212 
213 


597 

806 

32 015 


618 

827 
035 


639 

848 
056 


660 

869 
077 


681 

890 
098 


702 
911 
118 


723 
931 
139 


744 
952 
160 


765 
973 
181 


785 
994 
201 


21 
21 
21 


222 


243 


263 


284 


305 


325 


346 


366 


387 


408 


21 


426 
634 

838 


449 
654 

858 


469 
675 

879 


490 
695 
899 


510 
715 
919 


531 
736 
940 


552 
756 
960 


572 

777 
980 


593 

797 
*001 


613 

818 

*021 


20 
20 
20 


214 
215 
216 


33 041 
244 
445 


062 
264 
465 


082 
284 
486 


102 
304 

506 


122 

325 
526 


143 

345 
546 


163 
365 
566 


183 

385 
586 


203 

405 
606 


224 
425 
626 


20 
20 
20 


217 
218 
219 

220 

221 
222 
223 


646 

846 

34 044 


666 
866 
064 


686 

885 
084 


706 
905 
104 


726 
925 
124 


746 
945 
143 


766 
965 
163 


786 
985 
183 


806 

*005 

203 


826 

*025 

223 


20 
20 
20 


242 


262 


282 


301 


321 


341 


361 


380 


400 


420 


20 


439 
635 

830 


459 
655 
850 


479 
674 
869 


498 
: 694 

889 


518 
713 
908 


537 
733 

928 


557 
753 
947 


577 
772 
967 


596 
792 
986 


616 

811 

*005 


20 
19 

19 


224 
225 
226 


35 025 
218 
411 


044 
238 
430 


064 
257 
449 


083 
276 
468 


102 
295 

488 


122 
315 

507 


141 

1 334 
526 


160 
353 
545 


180 
372 
564 


199 
392 
583 


19 
19 
19 


227 
228 
229 


603 
793 

984 


622 

813 

*003 


641 

832 
*021 


660 
! 851 
*040 


679 

870 

*059 


698 

889 

*078 


717 
! 908 
i*097 


736 

927 

*116 


755 

946 

*135 


774 

965 

*154 


19 
19 
19 


N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


» 



COMMON LOGARITHMS OF NUMBERS 



N 





1 


2 


3 


4 


5 


6 


7 8 


9 


D 


230 

231 


36173 


192 


211 


229 


248 


267 


286 


305 


324 


342 


19 


361 


380 


399 


418 


436 


455 


474 


493 


511 


530 


19 


232 


549 


568 


586 


605 


624 


642 


661 


680 


698 


717 


19 


233 


736 


754 


773 


791 


810 


829 


847 


866 


884 


903 


19 


234 


922 


940 


959 


977 


996 


*014 


*033 


*051 


*070 


*088 


18 


235 


37 107 


125 


144 


162 


181 


199 


218 


236 


254 


273 


18 


23G 


291 


310 


328 


346 


365 


383 


401 


420 


438 


457 


18 


237 


475 


493 


511 


530 


548 


566 


585 


603 


621 


639 


18 


238 


658 


676 


694 


712 


731 


749 


767 


785 


803 


822 


18 


239 
240 
241 


840 


858 


876 


894 


912 


931 


949 


967 


985 


*003 


18 


38 021 


039 


057 


075 


093 


112 


130 


148 


166 


184 


18 


202 


220 


238 


256 


274 


292 


310 


328 


346 


364 


18 


242 


382 


399 


417 


435 


453 


471 


489 


507 


525 


543 


18 


243 


561 


578 


596 


614 


632 


650 


668 


686 


703 


721 


18 


244 


739 


757 


775 


792 


810 


828 


846 


863 


881 


899 


18 


245 


917 


934 


952 


970 


987 


*005 


*023 


*041 


*058 


*076 


18 


246 


39 094 


111 


129 


146 


164 


182 


199 


217 


235 


252 


18 


247 


270 


287 


305 


322 


340 


358 


375 


393 


410 


428 


18 


248 


445 


463 


480 


498 


515 


533 


550 


568 


585 


602 


18 


249 
250 
251 


620 


637 


655 


672 


690 


707 


724 


742 


759 


777 


17 


794 


811 


829 


846 


863 


881 


898 


915 


933 


950 


17 


967 


985 


*002 


*019 


*037 


*054 *071 


*0S8 


*106 


*123 


17 


252 


40140 


157 


175 


192 


209 


226 


243 


261 


278 


295 


17 


253 


312 


329 


346 


364 


381 


398 


415 


432 


449 


466 


17 


254 


483 


500 


518 


535 


552 


569 


586 


603 


620 


637 


17 


255 


654 


671 


688 


705 


722 


739 


756 


773 


790 


807 


17 


256 


824 


841 


858 


875 


892 


909 


926 


943 


960 


976 


17 


257 


993 


*010 


*027 


*044 


*061 


*078 *095 


*111 


*128 


*145 


17 


258 


41 162 


179 


196 


212 


229 


246 263 


280 


296 


313 


17 


259 
260 
261 


330 


347 


363 


380 


397 


414 430 


447 


464 


481 


17 


497 


514 


531 


547 


564 


581 


597 


614 


631 


647 


17 


664 


681 


697 


714 


731 


747 


764 


780 


797 


814 


17 


262 


830 


847 


863 


880 


896 


913 


929 


946 


963 


979 


16 


263 


996 


*012 


*029 


*045 


*062 


*078 


*095 


*111 


*127 


*144 


16 


264 


42160 


177 


193 


210 


226 


243 


259 


275 


292 


308 


16 


265 


325 


341 


357 


374 


390 


406 


423 


439 


455 


472 


16 


266 


488 


504 


521 


537 


553 


570 


586 


602 


619 


635 


16 


267 


651 


667 


684 


700 


716 


732 749 


765 


781 


797 


16 


268 


813 


830 


846 


862 


878 


894 1 911 


927 


943 


959 


16 


269 


975 


991 


*008 


*024 


*040 


*056 i*072 


*088 


*104 


*120 


16 


N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D 



COMMON LOGARITHMS OF NUMBERS 



N 


O 


1 


2 


3 


4 


5 


6 


7 


8 


9 


D 


270 

271 

272 
273 


43136 


152 


169 


185 


201 


217 


233 


249 


265 


281 


16 


297 
457 
616 


313 
473 

632 


329 
489 
648 


345 

505 
664 


361 
521 
680 


377 
537 
696 


393 
553 
712 


409 
569 

727 


425 

584 
743 


441 
600 

759 


16 
16 
16 


274 
275 
276 


775 

933 

44 091 


791 

949 
107 


807 
965 
122 


823 
981 
138 


838 
996 
154 


854 

*012 

170 


870 

*028 

185 


886 

*044 

201 


902 
*059 

217 


917 

*075 
232 


16 
16 
16 


277 
278 
279 

280 

281 
282 
283 


248 
404 
560 


264 
420 
576 


279 
436 
592 


295 

451 
607 


311 

467 
623 


326 

483 
638 


342 
498 
654 


358 
514 
669 


373 

529 

685 


389 
545 
700 


16 
16 
16 


716 


731 


747 


762 


778 


793 


809 


824 


840 


855 


15 


871 

45 025 

179 


886 
040 
194 


902 
056 
209 


917 
071 
225 


932 
086 
240 


948 
102 
255 


963 
117 
271 


979 
133 

286 


994 
148 
301 


*010 
163 
317 


15 
15 
15 


284 

285 
286 


332 

484 
637 


347 
500 
652 


362 
515 
667 


378 
530 
682 


393 
545 
697 


408 
561 
712 


423 
576 

728 


439 
591 
743 


454 

606 

758 


469 
621 

773 


15 
15 
15 


287 
288 
289 

290 

291 
292 
293 


788 

939 

46 090 


803 
954 
105 


818 
969 
120 


834 
984 
135 


849 

*000 

150 


864 

*015 

165 


879 

*030 

180 


894 

*045 

195 


909 

*060 

210 


924 
*075 

225 


15 

15 
15 


240 


255 


270 


285 


300 


315 


330 


345 


359 


374 


15 


389 
538 

687 


404 
553 
702 


419 
568 
716 


434 

583 
731 


449 
598 
746 


464 
613 
761 


479 
627 
776 


494 
642 

790 


509 
657 
805 


523 

672 
820 


15 
15 
15 


294 
295 
296 


835 

982 

47129 


850 
997 
144 


864 

*012 

159 


879 

*026 

173 


894 

*041 

188 


909 

*056 

202 


923 

*070 
217 


938 

*085 

232 


953 

*100 

246 


967 

*114 

261 


15 

15 
15 


297 
29S 
299 

300 

301 
302 
303 


276 
422 
567 


290 
436 

582 


305 
451 
596 


319 
465 
611 


334 

480 
625 


349 
494 
640 


363 
509 
654 


378 
524 
669 


392 
538 
683 


407 
553 
698 


15 
15 
15 


712 


727 


741 


756 


770 


784 


799 


813 


828 


842 


14 


857 

48 001 

144 


871 
015 
159 


8S5 
029 
173 


900 
044 

187 


914 

058 
202 


929 
073 
216 


943 

087 
230 


958 
101 
244 


972 
116 
259 


986 
130 
273 


14 
14 

14 


304 
305 
306 


287 
430 
572 


302 
444 
586 


316 
458 
601 


330 
473 
615 


344 

487 
629 


359 
501 
643' 


373 
515 
657 


387 
530 
671 


401 
544 
686 


416 

558 
700 


14 
14 

14 


307 
308 
309 


714 
855 
996 


728 

869 

*010 


742 

883 

*024 


756 

897 
*038 


770 

911 

*052 


785 

926 

*066 


799 

940 

*080 


813 

954 

*094 


827 

968 

*108 


841 

982 

*122 


14 

14 
14 


N 


O 


1 


2 


3 


4 


5 


6 


7 


8 


9 


D 



COMMON LOGARITHMS OF NUMBERS 



N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D 


310 

311 

312 
313 


49 136 


150 


164 


178 


192 


206 


220 


234 


248 


262 


14 


276 
415 
554 


290 
429 
568 


304 
443 

582 


318 
457 
596 


332 
471 
610 


346 
485 
624 


360 
499 
638 


374 
513 
651 


388 
527 
665 


402 
541 
679 


14 
14 
14 


314 

315 
316 


693 
831 
969 


707 
845 
982 


721 
859 
996 


734 

872 

*010 


748 

886 

*024 


762 

900 

*037 


776 

914 

*051 


790 

927 

*065 


803 

941 

*079 


817 

955 

*092 


14 
14 
14 


317 
318 
319 

320 

321 
323 
323 


50 106 
243 
379 


120 
256 
393 


133 

270 
406 


147 

264 
420 


161 

297 
433 


174 
311 

447 


188 
325 
461 


202 
338 
474 


215 
352 

4b8 


229 
365 
501 


14 
14 

14 


515 


529 


542 


556 


569 


583 


£96 


610 


623 


637 


14 


651 

786 
920 


664 

799 
934 


678 
813 
947 


691 
826 
961 


705 
840 
974 


718 
853 
987 


732 

866 

*001 


745 

880 

*014 


759 
893 

*028 


772 

907 

*041 


14 
13 

13 


324 

325 
326 


51055 

188 
322 


068 
202 
335 


081 
215 
348 


095 
2 8 
362 


108 
242 
375 


121 
255 

388 


135 
268 
402 


148 
282 
415 


162 
195 

428 


175 

308 
441 


13 
13 
13 


327 

328 
329 

330 

331 
332 
333 


455 

587 
720 


468 
601 
733 


481 
614 
746 


495 
627 
759 


£08 
640 

772 


521 

654 
786 


534 
667 
799 


.548 
680 
812 


561 
693 

825 


574 
706 

838 


13 
13 
13 


851 


865 


878 


891 


904 


917 


930 


943 


957 


970 


13 


983 

52 114 

244 


996 
127 

257 


*009 
140 
270 


*022 
153 

284 


*C35 
166 
297 


*048 
179 
310 


*C61 
1S2 
323 


*C75 
205 
336 


*088 
218 
349 


*101 
231 
362 


13 
13 
13 


334 
335 
336 


375 
504 
634 


388 
517 
647 


401 
530 
660 


414 
543 
673 


427 
5. e 6 
686 


440 
569 
699 


453 

582 
711 


466 

595 

724 


479 

608 
737 


492 
621 
750 


13 
13 
13 


337 
338 
339 

340 

341 
342 
343 


763 

892 

53 020 


776 
905 
033 


789 
917 
046 


8C2 
930 
058 


815 
943 
071 


827 
956 
084 


840 
969 
097 


853 
982 
110 


866 
994 
122 


879 

*007 

135 


13 
13 
13 


148 


161 


173 


186 


199 


212 


224 


237 

364 
4C1 
618 


250 


263 


13 


275 
403 
529 


288 
415 
542 


301 
4°8 
555 


314 
441 
567 


326 
453 

580 


339 
466 
593 


352 
479 
605 


377 
504 
631 


390 
517 
643 


13 
13 
13 


344 
345 
346 


656 

782 
908 


668 
794 
920 


681 
807 
933 


694 
820 
945 


706 
832 
958 


719 
845 
970 


732 
857 
983 


744 
870 
995 


757 

882 

*008 


769 

895 
*C20 


13 
13 
13 


347 
348 
349 


54 033 

158 
283 


045 
170 
295 


058 
183 
307 


070 
195 
320 


083 
208 
332 


095 
220 
345 


108 
233 
357 


120 
245 

370 


133 

258 
382 


145 

270 
394 


13 
12 

12 


N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D 



COMMON LOGARITHMS OF NUMBERS 



N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D 


350 

351 
352 
353 


407 


419 


432 


444 


456 


469 


481 


494 


506 


518 


12 


531 
654 

777 


543 

667 
790 


555 

679 

802 


568 
691 

814 


580 
704 
827 


593 
716 
839 


605 
728 
851 


617 
741 
864 


630 
753 

876 


642 

765 
888 


12 
12 
12 


354 
355 
356 


900 

55 023 

145 


913 
035 
157 


925 

047 
169 


937 
060 
182 


949 
072 
194 


962 
084 
206 


974 
096 
218 


986 
108 
230 


998 
121 
242 


*011 
133 
255 


12 
12 

12 


357 
358 
359 

360 

361 
362 
363 


267 
388 
509 


279 
400 
522 


291 
413 
534 


303 
425 
546 


315 
43? 
558 


328 
449 
570 


340 
461 

582 


352 
473 
594 


364 
485 
606 


376 
497 
618 


12 
12 
12 


630 


642 


654 


666 


678 


691 


703 


715 


727 


739 


12 


751 
871 
991 


763 

883 

*003 


775 

895 

*015 


787 

907 

*027 


799 

919 

*038 


811 

931 

*050 


823 
943 

*062 


835 

955 

*074 


847 

967 

*086 


859 
979 

*098 


12 
12 

12 


364 
365 
366 


56110 
229 
348 


122 
241 
360 


134 
253 
372 


146 

265 
384 


158 

277 
396 


170 

289 
407 


182 
301 
419 


194 
312 
431 


205 
324 
443 


217 
336 
455 


12 
12 

12 


367 
368 
369 

370 

371 
372 
373 


467 

585 
703 


478 
597 
714 


490 
608 
726 


502 
620 
738 


514 
632 

750 


526 
644 
761 


538 
656 
773 


549 

667 

785 


561 
679 

797 


573 
691 

808 


12 
12 
12 


820 


832 


844 


855 


867 


879 


891 


902 


914 


926 


12 


937 

57 054 

171 


949 
066 
183 


961 
078 
194 


972 

089 
206 


984 
101 
217 


996 
113 
229 


*008 
124 
241 


*019 
136 
252 


*031 
148 
264 


*043 
159 
276 


12 
12 
12 


374 
375 
376 


287 
403 
519 


299 
415 
530 


310 
426 
542 


322 
438 
553 


334 
449 
565 


345 
461 
576 


357 
473 

588 


368 
484 
600 


380 
496 
611 


392 
507 
623 


12 
12 
12 


377 
378 
379 

380 

381 

382 
383 


634 

749 
864 


646 
761 

875 


657 

772 
887 


669 

784 
898 


680 
795 
910 


692 
807 
921 


703 
818 
933 


715 
830 
944 


726 
841 
955 


738 
852 
967 


11 
11 
11 


978 


990 


*001 


*013 


*024 


*035 


*047 


*058 


*070 


*081 


11 


58 092 
206 
320 


104 
218 
331 


115 
229 
343 


127 
240 
354 


138 
252 
365 


149 
263 
377 


161 

274 

388 


172 

286 
399 


184 
297 
410 


195 

309 
422 


11 
11 
11 


384 
385 
386 


433 
546 
659 


444 
557 

670 


456 
569 

681 


467 
580 
692 


478 
591 
704 


490 
602 
715 


501 
614 
726 


512 
625 
737 


524 
636 
749 


535 

647 
760 


11 
11 
11 


387 
388 
389 


771 
883 
995 


782 

894 

*0C6 


794 

906 
*017 


805 

917 

*028 


816 

928 

*040 


827 

939 

*051 


838 

950 

*062 


850 

961 

*073 


861 

973 

*084 


872 

984 

*095 


11 
11 
11 


N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D 



IO 



COMMON LOGARITHMS OF NUMBERS 



N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D 


390 

391 
392 
393 


59106 


118 


129 


140 


151 


162 


173 


184 


195 


207 


11 


218 
329 
439 


229 
340 
450 


240 
351 
461 


251 

362 

472 


262 
373 
483 


273 
384 
494 


284 
395 
506 


295 
406 
517 


306 
417 

528 


318 
428 
539 


11 
11 
11 


394 
395 
396 


550 

660 

770 


561 
671 

780 


572 
682 
791 


583 
693 
802 


594 

704 
813 


605 
715 

824 


616 

726 
835 


627 
737 
846 


638 

748 

857 


649 
759 
868 


11 
11 
11 


397 
398 
399 

400 

401 
402 
403 


879 

988 

60 097 


890 
999 
108 


901 

*010 

119 


912 

*021 

130 


923 

*032 

141 


934 

*043 
152 


945 

*054 

163 


956 

*065 

173 


966 

*076 

184 


977 

*086 

195 


11 
11 
11 


206 


217 


228 


239 


249 


260 


271 


282 


293 


304 


11 


314 
423 
531 


325 
433 
541 


336 
444 
552 


347 
455 
563 


358 
466 
574 


369 

477 
584 


379 

487 
595 


390 
498 
606 


401 
509 
617 


412 
520 
627 


11 
11 
11 


404 
405 
406 


638 
746 
853 


649 
756 
863 


660 

767 
874 


670 

778 
bS5 


681 
788 
895 


692 
799 
906 


703 
810 
917 


713 

821 
927 


724 
831 
938 


735 
842 
949 


11 
11 
11 


407 
408 
409 

410 

411 
412 
413 


959 

61066 

172 


970 
077 
183 


981 
087 
194 


991 
098 
204 


*002 
109 

215 


*013 
119 

225 


*023 
130 
236 


*034 
140 

247 


*045 
151 

257 


*055 
162 
268 


11 
11 
11 


278 


289 


300 


310 


321 


331 


342 


352 


363 


374 


11 


384 
490 
595 


395 
500 
606 


405 
511 
616 


416 
521 
627 


426 
532 
637 


437 
542 
648 


448 
553 
658 


458 
563 
669 


469 
574 
679 


479 

584 
690 


11 
11 
11 


414 
415 
416 


700 
805 
909 


711 
815 
920 


721 
826 
930 


731 
836 
941 


742 
847 
951 


752 
857 
962 


763 

868 
972 


773 

878 
982 


784 
888 
993 


794 

899 

*003 


10 
10 
10 


417 
418 
419 

420 

421 
422 
423 


62 014 
118 
221 


024 

128 
232 


034 
138 
242 


045 
149 
252 


055 
159 
263 


066 
170 
273 


076 
180 

284 


086 
190 
294 


097 
201 
304 


107 
211 
315 


10 
10 
10 


325 


335 


346 


356 


366 


377 


387 


397 


408 


418 


10 


428 
531 
634 


439 
542 
644 


449 
552 
655 


459 
562 
665 


469 
572 
675 


480 
583 
685 


490 
593 
696 


500 
603 

706 


511 
613 
716 


521 
624 
726 


10 
10 

10 


424 
425 
426 


737 
839 
941 


747 
849 
951 


757 
859 
961 


767 
870 
972 


778 

880 
982 


788 
890 
992 


798 

900 

*002 


808 

910 

*012 


818 

921 

*022 


829 

931 

*033 


10 
10 
10 


427 
428 
429 


63 043 
144 
246 


053 
155 
256 


063 
165 
266 


073 
175 
276 


083 

185 
286 


094 
195 
296 


104 

205 
306 


114 

215 
317 


124 
225 
327 


134 

236 
337 


10 
10 
10 


N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D 



COMMON LOGARITHMS OF NUMBERS 



II 



N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D 




430 

431 
432 
433 


347 


357 


367 


377 


387 


397 


407 


417 


428 


438 


10 




448 
548 
649 


458 
558 
659 


468 
568 
669 


478 
579 
679 


488 
589 
689 


498 
599 
699 


508 
609 
709 


518 
619 
719 


528 
629 
729 


538 
639 
739 


10 
10 
10 




434 
435 
436 


749 
849 
949 


759 
859 
959 


769 
869 
969 


779 
879 
979 


789 
889 

988 


799 
899 
998 


809 

909 

*008 


819 

919 

*018 


829 

929 

*028 


839 

939 

*038 


10 
10 
10 




437 
438 
439 

440 

441 
442 
443 


64 048 
147 
246 


058 
157 
256 


068 
167 
266 


078 
177 
276 


088 
187 
286 


098 
197 
296 


108 
207 
306 


118 
217 
316 


128 
227 
326 


137 
237 
335 


10 
10 
10 




345 


355 


365 


375 


385 


395 


404 


414 


424 434 


10 




444 
542 
640 


454 
552 
650 


464 
562 
660 


473 
572 
670 


483 
582 
680 


493 
591 
689 


503 
601 
699 


513 

611 
709 


523 

621 
719 


532 
631 
729 


10 

10 
10 




444 
445 
446 


738 
836 
933 


748 
846 
943 


758 
856 
953 


768 
865 
963 


777 
875 
972 


787 
885 
982 


797 
895 
992 


807 

904 

*002 


816 

914 

*011 


826 

924 

*021 


10 
10 
10 




447 
448 
449 

450 

451 
452 
453 


65 031 

128 
225 


040 
137 
234 


050 
147 
244 


060' 

157 

254 


070 
167 
263 


079 
176 
273 


089 
186 

283 


099 
196 
292 


108 
205 
302 


118 
215 
312 


10 
10 

10 




321 


331 


341 


350 


360 


369 


379 


389 


398 


408 


10 




418 
514 
610 


427 
523 
619 


437 
533 
629 


447 
543 
639 


456 
552 

648 


466 
562 

658 


475 

571 
667 


485 
581 
677 


495 504 
591 600 
086 696 


10 
10 
10 




454 
455 
456 


706 
801 
896 


715 
811 
906 


725 
820 
916 


734 
830 
925 


744 
839 
935 


753 

849 
944 


763 

858 
954 


772 
868 
963 


782 
877 
973 


792 
887 
982 


9 
9 
9 




457 

458 
459 

460 

461 
462 
463 


992 

66 087 

181 


*001 
096 
191 


*011 
106 
200 


*020 
115 
210 


*030 
124 
219 


*039 
134 
229 


*049 
143 

238 


*058 
153 

247 


*068 
162 
257 


*077 
172 
266 


9 
9 
9 




276 


285 


295 


304 


314 


323 


332 


342 


351 


361 


9 




370 
464 
558 


380 
474 
567 


389 
483 

577 


398 
492 
586 


408 
502 
596 


417 
511 
605 


427 
521 
614 


436 
530 
624 


445 
539 
633 


455 

549 
642 


9 
9 
9 




464 
465 
466 


652 
745 
839 


661 
755 

848 


671 

764 
857 


680 
773 
367 


689 
783 

876 


699 

792 
885 


708 
801 
894 


717 

811 
904 


727 
820 
913 


736 
829 
922 


9 
9 

9 




467 
468 
469 


932 

67 025 

117 


941 
034 
127 


950 
043 
136 


960 
052 
145 


969 
062 
154 


978 
071 
164 


987 
080 
173 


997 
089 
182 


*006 
099 
191 


*015 
108 
201 


9 
9 
9 




N 





1 


2 


3 


4 5 


6 


7 


8 


9 


D 





12 



COMMON LOGARITHMS OF NUMBERS 



N 


O 


1 


2 


3 


4 


5 


6 


7 


8 


9 


D 


470 

471 


210 


219 


228 


237 


247 


256 


265 


274 


284 


293 


9 


302 


311 


321 


330 


339 


348 


357 


367 


376 


385 


9 


472 


394 


403 


413 


422 


431 


440 


449 


459 


468 


477 


9 


473 


486 


495 


504 


514 


523 


532 


541 


550 


560 


569 


9 


474 


578 


587 


596 


605 


614 


624 


633 


642 


651 


660 


9 


475 


669 


679 


688 


697 


706 


715 


724 


733 


742 


752 


9 


476 


761 


770 


779 


788 


797 


806 


815 


825 


834 


843 


9 


477 


853 


861 


870 


879 


888 


897 


906 


916 


925 


934 


9 


478 


943 


952 


961 


970 


979 


988 


997 


*006 


*015 


*024 


9 


479 
480 

481 


68 034 


043 


052 


061 


070 


079 


088 


097 


106 


115 


9 


124 


133 


142 


151 


160 


169 


178 


187 


196 


205 


9 


215 


224 


233 


242 


251 


260 


269 


278 


287 


296 


9 


482 


305 


314 


323 


332 


341 


350 


359 


368 


377 


386 


9 


483 


395 


404 


413 


422 


431 


440 


449 


458 


467 


476 


9 


484 


485 


494 


502 


511 


520 


529 


538 


547 


556 


565 


9 


485 


574 


583 


592 


601 


610 


619 


628 


637 


646 


655 


9 


486 


664 


673 


681 


690 


609 


708 


717 


726 


735 


744 


9 


487 


753 


763 


771 


780 


789 


797 


806 


815 


824 


833 


9 


488 


842 


851 


860 


869 


878 


886 


895 


904 


913 


922 


9 


489 
490 

491 


931 


940 


949 


958 


966 


975 


984 


993 


*002 


*011 


9 


69 020 


028 


037 


046 


055 


064 


073 


082 


090 


099 


9 


108 


117 


126 


135 


144 


152 


161 


170 


179 


188 


9 


492 


197 


205 


214 


223 


232 


241 


249 


258 


267 


276 


9 


493 


285 


294 


302 


311 


320 


329 


338 


346 


355 


364 


9 


494 


373 


381 


390 


399 


408 


417 


425 


434 


443 


452 


9 


495 


461 


469 


478 


487 


496 


504 


513 


522 


531 


539 


9 


496 


548 


557 


566 


574 


583 


592 


601 


609 


618 


627 


9 


497 


636 


644 


653 


662 


671 


679 


688 


697 


705 


714 


9 


498 


723 


732 


740 


749 


758 


767 


775 


784 


793 


801 


9 


499 
500 

501 


810 


819 


827 


836 


845 


854 


863 


871 


880 


888 


9 


897 


906 


914 


923 


932 


940 


949 


958 


966 


975 


9 


984 


992 


*001 


*010 


*018 


*027 


*036 


*044 


*053 


"062 


9 


502 


70 070 


079 


088 


096 


105 


114 


122 


131 


140 


148 


9 


503 


157 


165 


174 


183 


191 


200 


209 


217 


226 


234 


9 


504 


243 


252 


260 


269 


278 


286 


295 


303 


312 


321 


9 


505 


329 


338 


346 


355 


364 


372 


381 


389 


398 


406 


9 


506 


415 


424 


432 


441 


449 


458 


467 


475 


484 


492 


9 


507 


501 


509 


518 


526 


535 


544 


552 


561 


569 


578 


9 


508 


586 


595 


603 


612 


621 


629 


638 


646 


655 


663 


9 


509 


672 


680 


689 


697 


706 


714 


723 


731 


740 


749 


9 


N 


O 


1 


2 


3 


4 


5 


6 


7 


8 


9 


D 



COMMON LOGARITHMS OF NUMBERS 



13 



N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D 


510 

511 


757 


766 


774 


783 


791 


800 


808 


817 


825 


834 


9 


842 


851 


859 


868 


876 


885 


893 


902 


910 


919 


9 


512 


927 


935 


944 


952 


961 


969 


978 


986 


995 


*003 


9 


513 


71012 


020 


029 


037 


046 


054 


063 


071 


079 


088 


8 


514 


096 


105 


' 113 


122 


130 


139 


147 


155 


164 


172 


8 


515 


181 


189 


198 


206 


214 


223 


231 


240 


248 


257 


8 


516 


265 


273 


282 


290 


299 


307 


315 


324 


332 


341 


8 


517 


349 


357 


366 


374 


383 


391 


399 


408 


416 


425 


8 


518 


433 


441 


450 


458 


466 


475 


483 


492 


500 


508 


8 


519 
520 

521 


517 


525 


533 


542 


550 


559 


567 


575 


584 


592 


8 


600 


609 


617 


625 


634 


642 


650 


659 


667 


675 


8 


684 


692 


700 


709 


717 


725 


734 


742 


750 


759 


8 


522 


767 


775 


784 


792 


800 


809 


817 


825 


834 


842 


8 


523 


850 


858 


867 


875 


883 


892 


900 


908 


917 


925 


8 


524 


933 


941 


950 


958 


966 


975 


983 


991 


999 


*008 


8 


525 


72 016 


024 


032 


041 


049 


057 


066 


074 


082 


090 


8 


526 


099 


107 


115 


123 


132 


140 


148 


156 


165 


173 


8 


527 


181 


189 


198 


206 


214 


222 


230 


239 


247 


255 


8 


528 


263 


272 


280 


288 


296 


304 


313 


321 


329 


337 


8 


529 
530 

531 


346 


354 


362 


370 


378 


387 


395 


403 


411 


419 


8 


428 


436 


444 


452 


460 


469 


477 


485 


493 


501 


8 


509 


518 


526 


534 


542 


550 


558 


567 


575 


583 


8 


532 


591 


599 


607 


616 


624 


632 


640 


648 


656 


665 


8 


533 


673 


681 


689 


697 


705 


713 


722 


730 


738 


746 


8 


534 


754 


762 


770 


779 


787 


795 


803 


811 


819 


827 


8 


535 


835 


843 


852 


860 


868 


876 


884 


892 


900 


908 


8 


536 


916 


925 


933 


941 


949 


957 


965 


973 


981 


989 


8 


537 


997 


*006 


*014 


*022 


*030 


*038 


*046 


*054 


*062* 


F *070 


8 


538 


73 078 


086 


094 


102 


111 


119 


127 


135 


143 


151 


8 


539 
540 

541 


159 


167 


175 


183 


191 


199 


207 


215 


223 


231 


8 


239 


247 


255 


263 


272 


280 


28S 


296 


304 


312 


8 


320 


328 


336 


344 


352 


360 


368 


376 


384 


392 


8 


542 


400 


408 


416 


424 


432 


440 


448 


456 


464 


472 


8 


543 


480 


488 


496 


504 


512 


520 


528 


536 


544 


552 


8 


544 


560 


568 


576 


584 


592 


600 


608 


616 


624 


632 


8 


545 


640 


648 


656 


664 


672 


679 


687 


695 


703 


711 


8 


546 


719 


727 


735 


743 


751 


759 


767 


775 


783 


791 


8 


547 


799 


807 


815 


823 


830 


838 


846 


854 


862 


870 


8 


548 


878 


886 


894 


902 


910 


918 


926 


933 


941 


949 


8 


549 


957 


965 


973 


981 


989 


997 


*005 


*013 


*020 


*028 


8 


N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D 



14 



COMMON LOGARITHMS OF NUMBERS 



N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D 


550 

551 
552 
553 


74 036 


044 


052 


060 


068 


076 


084 


092 


099 


107 


8 


115 
194 
273 


123 

202 
280 


131 
210 

288 


139 
218 
296 


147 
225 
304 


155 

233 
312 


162 
241 
320 


170 
249 
327 


178 
257 
335 


186 
265 
343 


8 
8 
8 


554 
555 
556 


351 
429 
507 


359 
437 
515 


367 
445 
523 


374 
453 
531 


382 
461 
539 


390 
468 
547 


398 
476 
554 


406 

484 
562 


414 

492 
570 


421 

500 
578 


8 

8 
8 


557 
558 

559 

560 

561 
562 
563 


586 
663 
741 


593 

671 
749 


601 

679 
757 


609 
687 
764 


617 
695 

772 


624 

702 
780 


632 
710 

788 


640 
718 
796 


648 
726 
803 


656 
733 
811 


8 
8 
8 


819 


827 


834 


842 


850 


858 


865 


873 


881 


889 


8 


896 
974 

75 051 


904 
981 
059 


912 
989 
066 


920 
997 
074 


927 

*005 

082 


935 

*012 

089 


943 

*020 

097 


950 

*028 
105 


958 

*035 

113 


966 

*043 

120 


8 
8 
8 


564 
565 
566 


128 
205 

282 


136 
213 

289 


143 

220 
297 


151 

228 
305 


159 
236 
312 


166 
243 
320 


174 
251 

328 


182 
259 
335 


189 
266 
343 


197 
274 
351 


8 
8 
8 


567 
568 
569 

570 

571 
572 
573 


358 
435 
511 


366 
442 
519 


374 

450 
526 


381 
458 
534 


389 
465 
542 


397 
473 
549 


404 
481 
557 


412 
488 
565 


420 
496 

572 


427 
504 
580 


8 
8 
8 


587 


595 


603 


610 


618 


626 


633 


641 


648 


656 


8 


664 
740 
815 


671 

747 
823 


679 
755 

831 


686 
762 

838 


694 
770 
846 


702 
778 
853 


709 

785 
861 


717 
793 
868 


724 
800 
876 


732 
808 
884 


8 
8 
8 


574 
575 
576 


891 

967 

76 042 


899 
974 
050 


906 

982 
057 


914 
989 
065 


921 
997 
072 


929 

*005 

080 


937 

*012 
087 


944 

*020 
095 


952 

*027 
103 


959 

*035 

110 


8 
8 
8 


577 
578 
579 

580 

581 

582 
583 


118 
193 

268 


125 

200 

275 


133 

208 
283 


140 

215 
290 


148 
223 
298 


155 

230 
305 


163 

238 
313 


170 
245 
320 


178 
253 

328 


185 
260 
335 


8 
8 
8 


343 


350 


358 


365 


373 


380 


388 


395 


403 


410 


8 


418 
492 
567 


425 
500 
574 


433 
507 

582 


440 

515 
589 


448 
522 
597 


455 
530 
604 


462 
537 

612 


470 
545 
619 


477 
552 
626 


485 
559 
634 


7 
7 
7 


584 
585 
586 


641 
716 
790 


649 
723 
797 


656 

730 
805 


664 
738 
812 


671 
745 
819 


678 
753 

827 


686 
760 
834 


693 

768 
842 


701 

775 
849 


708 
782 
856 


7 
7 
7 


587 
588 
589 


864 

938 

77 012 


871 
945 
019 


879 
953 
026 


886 
960 
034 


893 
967 
041 


901 
975 
048 


908 
982 
056 


916 
989 
063 


923 

997 
070 


930 
*004 

078 


7 
7 
7 


N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D 



COMMON LOGARITHMS OF NUMBERS 



15 



N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D 


590 

591 
592 
593 


085 


093 


100 


107 


115 


122 


129 


137 


144 


151 


7 


159 
232 
305 


166 
240 
313 


173 
247 
320 


181 
254 

327 


188 
262 
335 


195 
269 
342 


203 
276 
349 


210 

283 
357 


217 
291 
364 


225 
298 
371 


7 
7 
7 


594 
595 
596 


379 
452 
525 


386 
459 

532 


393 
466 
539 


401 
474 
546 


408 
481 
554 


415 

488 
561 


422 
495 
568 


430 
503 
576 


437 
510 
583 


444 
517 
590 


7 
7 
7 


597 
598 
599 

600 

601 
602 
603 


597 
670 
743 


605 
677 
750 


612 

685 
757 


619 
692 
764 


627 
699 

772 


634 

706 
779 


641 
714 

786 


648 
721 
793 


656 

728 
801 


663 

735 
808 


7 
7 
7 


815 


822 


830 


837 


844 


851 


859 


866 


873 


880 


7 


887 

960 

78 032 


895 
967 
039 


902 
974 
046 


909 
981 
053 


916 

988 
061 


924 
996 

068 


931 

*003 

075 


938 
*010 

082 


945 

*017 

089 


952 

*025 

097 


7 
7 
7 


604 
605 
606 


104 
176 
247 


111 
183 
254 


118 
190 
262 


125 
197 
269 


132 
204 

276 


140 
211 
283 


147 

219 

290 


154 

226 

297 


161 

233 
305 


168 
240 
312 


7 
7 
7 


607 
608 
609 

610 

611 

612 
613 


319 

390 
462 


326 
398 
469 


333 
405 
476 


340 
412 
483 


347 
419 
490 


355 
426 

497 


362 
433 
504 


369 
440 
512 


376 
447 
519 


383 
455 
526 


7 
7 
7 


533 


540 


547 


554 


561 


569 


576 


583 


590 


597 


7 


604 
675 
746 


611 
682 
753 


618 
689 
760 


625 
696 
767 


633 

704 
774 


640 
711 

781 


647 

718 
789 


654 

725 
796 


661 

732 
803 


668 
739 
810 


7 
7 
7 


614 
615 
616 


817 

888 
958 


824 
895 
965 


831 
902 
972 


838 
909 
979 


845 
916 
986 


852 
923 
993 


859 

930 

*000 


866 

937 
*007 


873 

944 

*014 


880 

951 

*021 


7 
7 
7 


617 
618 
619 

620 

621 
622 
623 


79 029 
099 
169 


036 
106 
176 


043 
113 
183 


050 
120 
190 


057 
127 
197 


064 

134 
204 


071 
141 
211 


078 

148 
218 


085 
155 
225 


092 
162 
232 


7 
7 
7 


239 


246 


253 


260 


267 


274 


281 


288 


295 


302 


7 


309 
379 
449 


316 
386 
456 


323 

393 
463 


330 
400 
470 


337 
407 

477 


344 
414 
484 


351 
421 
491 


358 
428 
498 


365 
435 
505 


372 
442 
511 


7 

7 
7 


624 
625 
626 


518 

588 
657 


525 
595 
664 


532 
602 
671 


539 

609 
678 


546 
616 

685 


553 
623 
692 


560 
630 
699 


567 
637 
706 


574 
644 
713 


581 
650 
720 


7 
7 
7 


627 

628 
629 


727 
796 
865 


734 

803 
872 


741 
810 
879 


748 
817 
886 


754 

824 
893 


761 
831 
900 


768 

837 
906 


775 
844 
913 


782 
851 
920 


789 
858 
927 


7 
7 
7 


N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D 



i6 



COMMON LOGARITHMS OF NUMBERS 



N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


r> 


630 

631 


934 


941 


948 


955 


962 


969 


975 


982 


989 


996 


7 


80 003 


010 


017 


024 


030 


037 


044 


051 


058 


065 


7 


633 


072 


079 


085 


092 


099 


106 


113 


120 


127 


134 


7 


633 


140 


147 


154 


161 


168 


175 


182 


188 


195 


202 


7 


634 


209 


216 


223 


229 


236 


243 


250 


257 


264 


271 


7 


635 


277 


284 


291 


298 


305 


312 


318 


325 


332 


339 


7 


636 


346 


353 


359 


366 


373 


380 


387 


393 


400 


407 


7 


637 


414 


421 


428 


434 


441 


448 


455 


462 


468 


475 


7 


638 


482 


489 


496 


502 


509 


516 


523 


530 


536 


543 


7 


639 
640 

641 


550 


557 


564 


570 


577 


584 


591 


598 


604 


611 


7 


618 


625 


632 


638 


645 


652 


659 


665 


672 


679 

747 


7 


686 


693 


699 


706 


713 


720 


726 


too 


740 


7 


642 


754 


760 


767 


774 


781 


787 


794 


801 


808 


814 


7 


643 


821 


828 


835 


841 


848 


855 


862 


868 


875 


882' 


7 


644 


889 


895 


902 


909 


916 


922 


929 


936 


943 


949 


7 


645 


956 


963 


969 


976 


983 


990 


996 


*003 


*010 


*017 


7 


646 


81023 


030 


037 


043 


050 


057 


064 


070 


077 


084 


7 


647 


090 


097 


104 


111 


117 


124 


131 


137 


144 


151 


7 


648 


158 


164 


171 


178 


184 


191 


198 


204 


211 


218 


7 


649 
650 

651 


224 


231 


238 


245 


251 


258 


265 


271 


278 


285 


7 


291 


298 


305 


311 


318 


325 


331 


338 


345 


351 


7 


358 


365 


371 


378 


385 


391 


398 


405 


411 


418 


7 


652 


425 


431 


438 


445 


451 


458 


465 


471 


478 


485 


7 


653 


491 


498 


505 


511 


518 


525 


531 


538 


544 


551 


7 


654 


558 


564 


571 


578 


584 


591 


598 


604 


611 


617 


7 


655 


624 


631 


637 


644 


651 


657 


664 


671 


677 


684 


7 


656 


690 


697 


704 


710 


717 


723 


730 


737 


743 


750 


7 


657 


757 


763 


770 


776 


783 


790 


796 


803 


809 


816 


7 


658 


823 


829 


836 


842 


849 


856 


862 


869 


875 


882 


7 


659 
660 

661 


889 


895 


902 


908 


915 


921 


928 


935 


941 


948 


7 


954 


961 


968 


974 


981 


9S7 


994 


*000 


*007 


*014 


7 


82 020 


027 


033 


040 


046 


053 


060 


066 


073 


079 


7 


662 


086 


092 


099 


105 


112 


119 


125 


132 


138 


145 


7 


663 


151 


158 


164 


171 


178 


184 


191 


197 


204 


210 


7 


664 


217 


223 


230 


236 


243 


249 


256 


263 


269 


276 


7 


665 


282 


289 


295 


302 


308 


315 


321 


328 


334 


341 


7 


666 


347 


354 


360 


367 


373 


380 


387 


393 


400 


406 


7 


667 


413 


419 


426 


432 


439 


445 


452 


458 


465 


471 


7 


668 


478 


484 


491 


497 


504 


510 


517 


523 


530 


536 


7 


669 


543 


549 


556 


562 


569 


575 


582 


588 


595 


601 


7 


N 





1 


2 


3 


4 


5 


6 


n 


8 


9 


D 



COMMON LOGARITHMS OF NUMBERS 



17 



N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D 


670 


607 


614 


620 


627 


633 


640 


646 


653 


659 


666 


7 


671 
672 
673 


672 
737 
802 


679 
743 

808 


685 
750 
814 


692 

756 
821 


698 
763 
827 


705 
769 
834 


711 
776 
840 


718 
782 
847 


724 
189 
853 


730 
795 
860 


6 
6 
6 


674 
675 
676 


866 
930 
995 


872 

937 

*001 


879 

943 

*003 


885 

950 

*014 


892 
956 

*020 


898 
963 

*027 


905 

969 

*033 


911 

975 
*040 


918 

982 

*046 


924 

988 
*052 


6 

6 
6 


677 
678 
679 

680 

681 
682 
683 


83 059 
123 

187 


065 
129 
193 


072 
136 

200 


078 
142 
206 


085 
149 
213 


091 
155 
219 


097 
161 
225 


104 
168 
232 


110 
174 

238 


117 

181 
245 


6 
6 
6 


251 


257 


264 


270 


276 


283 


289 


296 


302 


308 


6 


315 

378 
442 


321 

385 
448 


327 
391 
455 


334 
398 
461 


340 
404 
467 


347 
410 
474 


353 

417 
480 


359 
423 

487 


366 
429 
493 


372 
436 
499 


6 
6 
6 


684 
685 
686 


506 
569 
632 


512 
575 
639 


518 

582 
645 


525 

588 
651 


531 
594 
658 


537 
601 
664 


544 
607 
670 


550 
613 
677 


556 
620 
683 


563 
626 
689 


6 
6 
6 


687 
688 
689 

690 

691 
692 
693 


696 
759 

822 


702 

765 
828 


708 
771 
835 


715 

778 
841 


721 
784 
847 


727 
790 
853 


734 
797 
860 


740 
803 
866 


746 
809 
872 


753 

816 
879 


6 
6 
6 


885 


891 


897 


904 


910 


916 


923 


929 


935 


942 


6 


948 

84 011 

073 


954 
017 
080 


960 
023 
086 


967 
029 

092 


973 
036 

098 


979 
042 
105 


985 
048 
111 


992 
055 
117 


998 
061 
123 


*004 
067 
130 


6 
6 
6 


694 
695 
696 


136 
198 
261 


142 

205 
267 


148 
211 
273 


155 
217 
280 


161 

223 
286 


167 
230 
292 


173 

236 
298 


180 
242 
305 


186 
248 
311 


192 

255 
317 


6 
6 
6 


697 
698 
699 

700 

701 
702 
703 


323 
386 

448 


330 
392 
454 


336 
398 
460 


342 
404 
466 


348 

410 
473 


354 
417 
479 


361 
423 

485 


367 
429 
491 


373 

435 
497 


379 
442 
504 


6 

6 
6 


510 


516 


522 


528 


535 


541 


547 


553 


559 


566 


6 


572 
634 
696 


578 
640 

702 


584 
646 

708 


590 
652 

714 


597 
658 
720 


603 
665 
726 


609 
671 
733 


615 

677 
739 


621 
683 
745 


628 
689 
751 


6 
6 
6 


704 
705 
706 


757 
819 

880 


763 

825 
887 


770 
831 
893 


776 

837 
899 


782 
844 
905 


788 
850 
911 


794 
856 
917 


800 
862 
924 


807 
868 
930 


813 

874 
936 


6 
6 
6 


707 
708 
709 


942 

85 003 
065 


948 
009 
071 


954 
016 

077 


960 
022 
083 


967 
028 
089 


973 

034 
095 


979 
040 
101 


985 
046 
107 


991 
052 
114 


997 
058 
120 


6 
6 
6 


N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D 



i8 



COMMON LOGARITHMS OF NUMBERS 



N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D 


710 

711 


126 


132 


138 


144 


150 


156 


163 


169 


175 


181 


6 


187 


193 


199 


205 


211 


217 


224 


230 


236 


242 


6 


712 


248 


254 


260 


266 


272 


278 


285 


291 


297 


303 


6 


713 


309 


315 


321 


327 


333 


339 


345 


352 


358 


364 


6 


714 


370 


376 


382 


388 


394 


400 


406 


412 


418 


425 


6 


715 


431 


437 


443 


449 


455 


461 


467 


473 


479 


485 


6 


716 


491 


497 


503 


509 


516 


522 


528 


534 


540 


546 


6 


717 


552 


558 


564 


570 


576 


582 


588 


594 


600 


606 


6 


718 


612 


618 


625 


631 


637 


643 


649 


655 


661 


667 


6 


719 
720 

721 


673 


679 


685 


691 


697 


703 


709 


715 


721 


727 


6 


733 


739 


745 


751 


757 


763 


769 


775 


781 


788 


6 


794 


800 


806 


812 


818 


824 


830 


836 


842 


848 


6 


722 


854 


860 


866 


872 


878 


884 


890 


890 


902 


908 


6 


723 


914 


920 


926 


932 


938 


944 


950 


956 


962 


968 


6 


724 


974 


980 


986 


992 


998 


-004 


*010 


*016 


*022 


*028 


6 


725 


86 034 


040 


046 


052 


058 


064 


070 


076 


082 


088 


6 


726 


094 


100 


106 


112 


118 


124 


130 


136 


141 


147 


6 


727 


153 


159 


165 


171 


177 


183 


189 


195 


201 


207 


6 


728 


213 


219 


225 


231 


237 


243 


249 


255 


261 


267 


6 


729 
730 
731 


273 


279 


285 


291 


297 


303 


308 


314 


320 


326 


6 


332 


338 


344 


350 


356 


362 


368 


374 


380 


386 


6 


392 


398 


404 


410 


415 


421 


427 


433 


439 


445 


6 


732 


451 


457 


463 


469 


475 


481 


4S7 


493 


499 


504 


6 


733 


510 


516 


522 


528 


534 


540 


546 


552 


558 


564 


6 


734 


570 


576 


581 


5S7 


593 


599 


605 


611 


617 


623 


6 


735 


629 


635 


641 


646 


652 


658 


664 


670 


676 


682 


6 


736 


688 


694 


700 


705 


711 


717 


723 


729 


735 


741 


6 


737 


747 


753 


759 


764 


770 


776 


782 


788 


794 


800 


6 


738 


806 


812 


817 


823 


829 


835 


841 


847 


853 


859 


6 


739 
740 

741 


864 


870 


876 


882 


888 


894 


900 


906 


911 


917 


6 


923 


929 


935 


941 


947 


953 


958 


964 


970 


976 


6 


982 


988 


994 


999 


*005 


*011 


*017 


*023 


*029 


*035 


6 


742 


87 040 


046 


052 


058 


064 


070 


075 


081 


087 


093 


6 


743 


099 


105 


111 


116 


122 


128 


134 


140 


146 


151 


6 


744 


157 


163 


169 


175 


181 


186 


192 


198 


204 


210 


6 


745 


216 


221 


227 


233 


239 


245 


251 


256 


262 


268 


6 


746 


274 


280 


286 


291 


297 


303 


309 


315 


320 


326 


6 


747 


332 


338 


344 


349 


355 


361 


367 


373 


379 


384 


6 


748 


390 


396 


402 


408 


413 


419 


425 


431 


437 


442 


6 


749 


448 


454 


460 


466 


471 


477 


483 


489 


495 


500 


6 


N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D 



COMMON LOGARITHMS OF NUMBERS 



J 9 



N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D 


750 

751 

752 
753 


506 


512 


518 


523 


529 


535 


541 


547 


552 


558 


6 


564 
622 
679 


570 

628 
685 


576 
633 
691 


581 
639 
697 


587 
645 
703 


593 
651 

708 


599 
656 
714 


604 
662 
720 


610 
668 
726 


616 
674 
731 


6 
6 
6 


754 
755 

756 


737 
795 
852 


743 

800 

858 


749 
806 
864 


754 
812 
869 


760 

, 818 
875 


766 
823 

881 


772 
829 
887 


777 
835 
892 


783 
841 
898 


789 
846 
904 


6 
6 
6 


757 
758 
759 

760 

761 

762 
763 


910 

967 
88 024 


915 
973 
030 


921 

978 
036 


927 
984 
041 


933 
990 
047 


938 
996 
053 


944 

*001 
058 


950 

*007 
064 


955 

*013 

070 


961 

*018 
076 


6 
6 
6 


0S1 


087 


093 


098 


104 


110 


116 


121 


127 


133 


6 


138 
195 
252 


144 

201 
258 


150 

207 
264 


156 
213 
270 


161 
218 
275 


167 
224 
281 


173 
230 

287 


178 

235 
292 


184 
241 
298 


190 
247 
304 


6 
6 
6 


764 
765 
766 


309 
366 
423 


315 

372 
429 


321 
377 
434 


326 
383 
440 


332 
389 
446 


338 
395 
451 


343 

400 
457 


349 
406 
463 


355 
412 

468 


360 
417 
474 


6 
6 
6 


767 
768 
769 

770 

771 

772 
773 


480 
536 
593 


485 
542 

598 


491 
547 

604 


497 
553 
610 


502 
559 
615 


508 
564 
621 


513 

570 
627 


519 
576 
632 


525 

581 
638 


530 

587 
643 


6 
6 
6 


649 


655 


660 


666 


672 


677 


683 


689 


694 


700 


6 


705 
762 
818 


711 

767 
824 


717 

773 
829 


722 
779 
835 


728 
784 
840 


734 

790 
846 


739 
795 
852 


745 

801 
857 


750 

807 
863 


756 
812 
868 


6 
6 
6 


774 
775 
776 


874 
930 
986 


880 
936 
992 


885 
941 
997 


891 

947 

*003 


897 

953 

*009 


902 

958 

*014 


908 

964 

*020 


913 

969 

*025 


919 

975 

*031 


925 

981 

*037 


6 
6 
6 


777 
778 
779 

780 

781 

782 
783 


89 042 
098 
154 


048 
104 
159 


053 
109 
165 


059 
115 
170 


064 
120 
176 


070 
126 

182 


076 
131 
187 


081 
137 
193 


087 
143 
198 


092 

148 
204 


6 
6 
6 


209 


215 


221 


226 


232 


237 


243 


248 


254 


260 


6 


265 
321 
376 


271 
326 
382 


276 
332 

3S7 


282 
337 
393 


287 
343 
398 


293 
348 
404 


298 
354 
409 


304 
360 
415 


310 
365 
421 


315 
371 
426 


6 
6 
6 


784 
785 
786 


432 

487 
542 


437 

492 

548 


443 
498 
553 


448 
504 
559 


454 
509 
564 


459 
515 
570 


465 
520 
575 


470 
526 
581 


476 
531 
586 


481 
537 
592 


6 
6 
6 


787 
788 
789 


597 
653 

708 


603 
658 
713 


609 
664 
719 


614 
669 

724 


620 
675 
730 


625 
680 
735 


631 
686 
741 


636 
691 
746 


642 
697 
752 


647 
702 
757 


6 
6 
6 


N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D 



20 



COMMON LOGARITHMS OF NUMBERS 



N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D 


790 

791 

792 
793 


763 


768 


774 


779 


785 


790 


796 


801 


807 


812 


5 


818 
873 
927 


823 
878 
933 


829 

883 
938 


834 
889 
944 


840 
894 
949 


845 
900 
955 


851 
905 
960 


856 
911 
966 


862 
916 
971 


867 
922 

977 


5 
5 
5 


794 

795 
796 


982 

90 037 

091 


988 
042 
097 


993 
048 
102 


998 
053 
108 


*004 
059 
113 


*009 
064 
119 


*015 
069 
124 


*020 
075 
129 


*026 
080 
135 


*031 
086 
140 


5 
5 
5 


797 
798 
799 

800 

801 
802 
803 


146 
200 
255 


151 

206 
260 


157 
211 
266 


162 
217 
271 


168 

222 
276 


173 

227 
282 


179 
233 

287 


184 

238 
293 


189 
244 

298 


195 
249 
304 


5 
5 
5 


309 


314 


320 


325 


331 


336 


342 


347 


352 


358 


5 


363 

417 
472 


369 
423 

477 


374 

428 
482 


380 
434 

488 


385 
439 
493 


390 
445 
499 


396 
450 
504 


401 
455 
509 


407 
461 
515 


412 
466 
520 


5 
5 
5 


804 
805 
806 


526 
580 
634 


531 

585 
639 


536 
590 
644 


542 
596 
650 


547 
601 
655 


553 

607 
660 


558 
612 
666 


563 
617 
671 


569 
623 
677 


574 

628 
682 


5 
5 
5 


807 
808 
809 

810 

811 

812 
813 


687 
741 
795 


693 

747 
800 


698 
752 

806 


703 

757 
811 


709 
763 
816 


714 

768 
822 


720 

773 
827 


725 
779 
832 


730 

784 
838 


736 
789 
843 


5 
5 
5 


849 


854 


859 


865 


870 


875 


881 


886 


891 


897 


5 


902 

956 

91009 


907 
961 
014 


913 
966 
020 


918 
972 
025 


924 
977 
030 


929 
982 
036 


934 
988 
041 


940 
993 
046 


945 

998 
052 


950 

*004 

057 


5 
5 
5 


814 

815 
816 


062 
116 
169 


068 
121 
174 


073 
126 

180 


078 
132 

185 


084 
137 
190 


089 
142 
196 


094 
148 
201 


100 
153 
206 


105 
158 
212 


110 
164 

217 


5 
5 
5 


817 
818 
819 

820 

821 
822 
823 


222 

275 
328 


228 
281 
334 


233 
286 
339 


238 
291 
344 


243 
297 
350 


249 
302 
355 


254 
307 
360 


259 
312 
365 


265 
318 
371 


270 
323 
376 


5 
5 
5 


381 


387 


392 


397 


403 


408 


413 


418 


424 


429 


5 


434 

487 
540 


440 
492 
545 


445 

498 
551 


450 
503 
556 


455 

508 
561 


461 
514 
566 


466 
519 
572 


471 
524 

577 


477 
529 

582 


482 
535 
587 


5 
5 
5 


824 
825 
826 


593 
645 
698 


598 
651 
703 


603 
656 
709 


609 
661 
714 


614 
666 
719 


619 

672 

724 


624 
677 
730 


630 

682 
735 


635 
687 
740 


640 
693 
745 


5 
5 
5 


827 
828 
829 


751 
803 
855 


756 
808 
861 


761 
814 
866 


766 
819 
871 


772 
824 
876 


777 
829 
882 


782 
834 

887 


787 
840 
892 


793 

845 
897 


798 
850 
903 


5 
5 
5 


N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D 



COMMON LOGARITHMS OF NUMBERS 



21 



N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D 


830 

831 

832 
833 


908 


913 


918 


924 


929 


934 


939 


944 


950 


955 


5 


960 

92 012 

065 


965 
018 
070 


971 
023 
075 


976 
028 
080 


981 
033 
085 


986 
038 
091 


991 
044 
096 


997 
049 
101 


*002 
054 
106 


*007 
059 
111 


5 
5 
5 


834 
835 
836 


117 

169 
221 


122 

174 
226 


127 
479 
231 


132 

134 
236 


137 

189 
241 


143 

195 
247 


148 
200 
252 


153 

205 
257 


158 

210 
262 


163 

215 
267 


5 

5 
5 


837 
838 
839 

840 

841 
842 
843 


273 
324 
376 


278 
330 
381 


283 

335 

387 


288 
340 
392 


293 
345 

397 


298 
350 
402 


304 
355 

407 


309 
361 
412 


314 
366 
418 


319 
371 
423 


5 
5 
5 


428 


433 


438 


443 


449 


454 


459 


464 


469 


474 


5 

5 
5 
5 


480 
531 

583 


485 
536 

588 


490 
542 
593 


495 
547 
598 


500 
552 
603 


505 
557 
609 


511 
562 
614 


516 

567 
619 


521 

572 
624 


526 
578 
629 


844 
845 
846 


634 

686 
737 


639 
691 
742 


645 

696 
747 


650 

701 
752 


655 
706 

758 


660 
711 
763 


665 
716 
768 


670 
722 
773 


675 

727 
778 


681 

732 
783 


5 
5 
5 


847 
848 
849 

850 

851 
852 
80S 


788 
840 
891 


793 

845 
896 


799 
850 
901 


804 
855 
906 


809 
860 
911 


814 
865 
916 


819 
870 
921 


824 
875 
927 


829 
881 
932 


834 
886 
937 


5 
5 
5 


942 


947 


952 


957 


962 


967 


973 


978 


983 


988 


5 


993 

93 044 
095 


998 
049 
100 


*003 
054 
105 


*008 
059 
110 


*013 
064 
115 


*018 
069 
120 


*024 
075 
125 


*029 
080 
131 


*034 
085 
136 


*039 
090 
141 


5 
5 
5 


854 
a55 
856 


146 
197 
247 


151 

202 
252 


156 

207 
258 


161 

212 
263 


166 

217 
268 


171 

222 
273 


176 
227 
278 


181 
232 

283 


186 
237 

288 


192 

242 
293 


5 
5 
5 


857 
858 
859 

860 

861 
862 
863 


298 
349 
399 


303 
354 
404 


308 
359 
409 


313 

364 
414 


318 
369 
420 


323 
374 
425 


328 
379 
430 


334 
384 
435 


339 
389 
440 


344 
394 
445 


5 
5 
5 


450 


455" 


460 


465 


470 


475 


480 


485 


490 


495 


5 


500 
551 
601 


505 
556 
606 


510 
561 
611 


515 
566 
616 


520 
571 
621 


526 
576 
626 


531 
581 
631 


536 
586 
636 


541 
591 
641 


546 
596 
646 


5 

5 
5 


864 
865 
866 


651 
702 
752 


656 

707 
757 


661 

712 
762 


666 
717 
767 


671 
722 

772 


676 

727 
777 


682 
732 

782 


687 
737 

787 


692 
742 
792 


697 

747 
797 


5 
5 
5 


867 
868 
869 


802 
852 
902 


807 
857 
907 


812 
862 
912 


817 
867 
917 


822 
872 
922 


827 
877 
927 


832 
882 
932 


837 
887 
937 


842 
892 
942 


847 
897 
947 


5 
5 
5 


N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D 



22 



COMMON LOGARITHMS OF NUMBERS 



N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D 


870 


952 


957 


962 


967 


972 


977 


982 


987 


992 


997 


5 


871 
872 
873 


94 002 
052 
101 


007 
057 
106 


012 
062 
111 


017 
067 
116 


022 
072 
121 


027 
077 
126 


032 
082 
131 


037 
086 
136 


042 
091 
141 


047 
096 
146 


5 
5 

5 


874 
875 
876 


151 

201 
250 


156 
206 
255 


161 
211 
260 


166 
216 
265 


171 

221 
270 


176 
226 

275 


181 
231 

280 


186 
236 

285 


191 

240 
290 


196 
245 
295 


5 
5 
5 


877 
878 
879 

880 

881 
882 
883 


300 
349 
399 


305 
354 
404 


310 
359 
409 


315 
364 
414 


320 
369 
419 


325 
374 
424 


330 

379 
429 


335 
384 

433 


340 

389 
438 


345 
394 

443 


5 
5 
5 


448 


453 


458 


463 


468 


473 


478 


483 


488 


493 


5 


498 
547 
596 


503 
552 
601 


507 
557 
606 


512 
562 
611 


517 
567 
616 


522 
571 
621 


527 
576 
626 


532 
581 
630 


537 
586 
635 


542 
591 
640 


5 
5 
5 


884 
885 
886 


645 
694 
743 


650 
699 

748 


655 
704 
753 


660 

709 

758 


665 

714 
763 


670 
719 

768 


675 

724 
773 


680 
729 

778 


685 
734 

783 


689 
738 
787 


5 
5 
5 


887 
888 
889 

890 

891 
892 
893 


792 
841 
890 


797 
846 

895 


802 
851 
900 


807 
856 
905 


812 
861 
910 


817 
866 
915 


822 
871 
919 


827 
876 
924 


832 

880 
929 


836 

885 
934 


5 
5 
5 


939 


944 


949 


954 


959 


963 


968 


973 


978 


983 


5 


988 

95 036 

085 


993 
041 
090 


998 
046 
095 


*002 
051 
100 


*007 
056 
105 


*012 
061 
109 


*017 
066 
114 


*022 
071 
119 


*027 
075 
124 


*032 
080 
129 


5 
5 
5 


894 

895 
896 


134 

182 
231 


139 

187 
236 


143 

192 
240 


148 
197 
245 


153 

202 
250 


158 
207 
255 


163 
211 
260 


168 
216 
265 


173 
221 
270 


177 
226 
274 


5 
5 
5 


897 
898 
899 

900 

901 
902 
903 


279 

328 
376 


284 
332 
381 


289 
337 
386 


294 
342 
390 


299 
347 
395 


303 

352 
400 


308 
357 
405 


313 
361 
410 


318 
366 
415 


323 
371 
419 


5 
5 
5 


424 


429 


434 


439 


444 


448 


453 


458 


463 


468 


5 


472 
521 
569 


477 
525 
574 


482 
530 

578 


487 
535 
583 


492 

540 

588 


497 
545 
593 


501 
550 
598 


506 
554 
602 


511 
559 
607 


516 
564 
612 


5 
5 
5 


904 
905 
906 


617 
665 
713 


622 
670 
718 


626 
674 

722 


631 
679 

727 


636 

684 
732 


641 
689 
737 


646 
694 
742 


650 
698 
746 


655 
703 
751 


660 
708 
756 


5 
5 
5 


907 
908 
909 


761 

809 
856 


766 
813 
861 


770 
818 
866 


775 
823 
871 


780 
828 
875 


785 
832 
880 


789 
837 
885 


794 
842 
890 


799 
847 
895 


804 
852 
899 


5 
5 
5 


N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D 



COMMON LOGARITHMS OF NUMBERS 



23 



N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D 


910 

911 
912 
913 


904 


909 


914 


918 


923 


928 


933 


938 


942 


947 


5 


952 

999 
96 047 


957 

*004 

052 


961 

*009 

057 


966 

*014 

061 


971 

*019 

066 


976 

*023 

071 


980 

*028 

076 


985 

*033 

080 


990 

*038 
085 


995 

*042 

090 


5 

5 
5 


914 

915 
916 


095 
142 
190 


099 
147 
194 


104 
152 
199 


109 
156 
204 


114 
161 
209 


118 
166 
213 


123 
171 

218 


128 
175 
223 


133 

180 
227 


137 
185 
232 


5 
5 
5 


917 
918 
919 

920 

921 
922 
923 


237 
284 
332 


242 
289 
336 


246 
294 
341 


251 
298 
346 


256 
303 
350 


261 
308 
355 


265 
313 
360 


270 
317 
365 


275 
322 
369 


280 
327 
374 


5 
5 
5 


379 


384 


388 


393 


398 


402 


407 


412 


417 


421 


5 


426 
473 
520 


431 
478 
525 


435 
483 
530 


440 
487 
534 


445 
492 
539 


450 
497 
544 


454 
501 
548 


459 
506 
553 


464 
511 
558 


468 
515 
562 


5 
5 
5 


924 
925 
926 


567 
614 
661 


572 
619 
666 


577 
624 
670 


581 
628 
675 


586 
633 

680 


591 

638 
685 


595 
642 
689 


600 

647 
694 


605 
652 
699 


609 
656 
703 


5 
5 
5 


927 
628 
929 

930 

931 
932 
933 


708 
755 

802 


713 
759 
806 


717 
764 
811 


722 
769 
816 


727 
774 
820 


731 

778 
825 


736 

783 
830 


741 

788 
834 


745 

792 
839 


750 

797 

844 


5 
5 
5 


848 


853 


858 


862 


867 


872 


876 


881 


886 


890 


5 


895 
942 

988 


900 
946 
993 


904 
951 
997 


909 

956 

*002 


914 

960 

*007 


918 

965 

*011 


923 

970 

*016 


928 

974 

*021 


932 

979 

*025 


937 

984 

*030 


5 
5 
5 


934 

935 
936 


97 035 
081 
128 


C39 
086 
132 


044 
090 
137 


049 
095 
142 


053 
100 
146 


058 
104 
151 


063 
109 
155 


067 
114 
160 


072 
118 
165 


077 
123 
169 


5 

5 
5 


937 
938 
939 

940 

941 
942 
943 


174 

220 
267 


179 

225 
271 


183 
230 
276 


188 
234 
280 


192 
239 

285 


197 
243 
290 


202 
248 
294 


206 
253 

299 


211 
257 
304 


216 
262 
308 


5 
5 
5 


313 


317 


322 


327 


331 


336 


340 


345 


350 


354 


5 


359 
405 
451 


364 
410 
456 


368 
414 
460 


373 
419 
465 


377 
424 
470 


382 
428 
474 


387 
433 
479 


391 
437 
483 


396 
442 

488 


400 
447 
493 


5 
5 
5 


944 
945 
946 


497 
543 
589 


502 
548 
594 


506 
552 
598 


511 

557 
603 


516 
562 

607 


520 
566 
612 


525 
571 
617 


529 
575 
621 


534 
580 
626 


539 

585 
630 


5 
5 
5 


947 
948 
949 


635 

681 
727 


640 
685 
731 


644 
690 
736 


649 

695 
740 


653 
699 
745 


658 
704 
749 


663 
708 
754 


667 
713 
759 


672 
717 
763 


676 
722 
768 


5 
5 
5 


N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D 



24 



COMMON LOGARITHMS OF NUMBERS 



N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D 


950 

951 


772 


777 


782 


786 


791 


795 


800 


804 


809 


813 


5 


818 


823 


827 


832 


836 


841 


845 


850 


855 


859 


5 


952 


864 


868 


873 


877 


882 


886 


891 


896 


900 


905 


5 


953 


909 


914 


918 


923 


928 


932 


937 


941 


946 


950 


5 


954 


955 


959 


964 


968 


973 


978 


982 


987 


991 


996 


5 


955 


98 000 


005 


009 


014 


019 


023 


028 


032 


037 


041 


5 


956 


046 


050 


055 


059 


064 


068 


073 


078 


082 


087 


5 


957 


091 


096 


100 


105 


109 


114 


118 


123 


127 


132 


5 


958 


137 


141 


146 


150 


155 


159 


164 


168 


173 


177 


5 


959 
960 

961 


182 


186 


191 


195 


200 


204 


209 


214 


218 


223 


5 


227 


232 


236 


241 


245 


250 


254 


259 


263 


268 


5 


272 


277 


281 


286 


290 


295 


299 


304 


308 


313 


5 


962 


318 


322 


327 


331 


336 


340 


345 


349 


354 


358 


5 


963 


363 


367 


372 


376 


381 


385 


390 


394 


399 


403 


5 


964 


408 


412 


417 


421 


426 


430 


435 


439 


444 


448 


5 


965 


453 


457 


462 


466 


471 


475 


480 


484 


489 


493 


4 


966 


498 


502 


507 


511 


516 


520 


525 


529 


534 


538 


4 


967 


543 


547 


552 


556 


561 


565 


570 


574 


579 


583 


4 


968 


588 


592 


597 


601 


605 


610 


614 


619 


623 


628 


4 


969 
970 

971 


632 


637 


641 


646 


650 


655 


659 


664 


668 


673 


4 


677 


682 


686 


691 


695 


700 


704 


709 


713 


717 


4 


722 


726 


731 


735 


740 


744 


749 


753 


758 


762 


4 


972 


767 


771 


776 


780 


784 


789 


793 


798 


802 


807 


4 


973 


811 


816 


820 


825 


829 


834 


838 


843 


847 


851 


4 


974 


856 


860 


865 


869 


874 


878 


883 


887 


892 


896 


4 


975 


900 


905 


909 


914 


918 


923 


927 


932 


936 


941 


4 


976 


945 


949 


954 


958 


963 


967 


972 


976 


981 


985 


4 


977 


989 


994 


998 


*003 


*007 


*012 


*016 


*021 


*025 


*029 


4 


978 


99 034 


038 


043 


047 


052 


056 


061 


065 


069 


074 


4 


979 
980 

981 


078 


083 


087 


092 


096 


100 


105 


109 


114 


118 


4 


123 


127 


131 


136 


140 


145 


149 


154 


158 


162 


4 


167 


171 


176 


180 


185 


189 


193 


198 


202 


207 


4 


982 


211 


216 


220 


224 


229 


233 


238 


242 


247 


251 


4 


983 


255 


260 


264 


269 


273 


277 


282 


286 


291 


295 


4 


984 


300 


304 


308 


313 


317 


322 


326 


330 


335 


339 


4 


985 


344 


348 


352 


357 


361 


366 


370 


374 


379 


383 


4 


986 


388 


392 


396 


401 


405 


410 


414 


419 


423 


427 


4 


9S7 


432 


436 


441 


445 


449 


454 


458 


463 


467 


471 


4 


988 


476 


480 


484 


489 


493 


498 


502 


506 


511 


515 


4 


989 


520 


524 


528 


533 


537 


542 


546 


550 


555 


559 


4 


N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D 



COMMON LOGARITHMS OF NUMBERS 



25 



N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D 


990 

991 


564 


568 


572 


577 


581 


585 


590 


594 


599 


603 


4 


60? 


612 


616 


621 


625 


629 


634 


638 


642 


647 


4 


992 


651 


656 


660 


664 


669 


673 


677 


682 


686 


691 


4 


993 


695 


699 


704 


708 


712 


717 


721 


726 


730 


734 


4 


994 


739 


743 


747 


752 


756 


760 


765 


769 


774 


778 


4 


995 


782 


787 


791 


795 


800 


804 


808 


813 


817 


822 


4 


996 


826 


830 


835 


839 


843 


848 


852 


856 


861 


865 


4 


997 


870 


874 


878 


883 


887 


891 


896 


900 


904 


909 


4 


998 


913 


917 


922 


926 


930 


935 


939 


944 


948 


952 


4 


999 


957 


961 


965 


970 


974 


978 


983 


987 


991 


996 


4 


N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D 



TABLE II. 
FIVE-PLACE LOGARITHMS 



OF THE 



SINE, COSINE, TANGENT, AND COTANGENT 



FOR 



EACH MINUTE FROM 0° TO 90°. 



26 



o c 



27 



t 


L. Sin. 


L. Tan. 


L. Cot. 


L. Cos. 






1 


CO 


CO 


CO 


0.00 000 


60 

59 


6.46 373 


6.46 373 


3.53 627 


0.00 000 


2 


6.76 476 


6.76 476 


3.23 524 


0.00 000 


58 


3 


6.94 085 


6.94 085 


3.05 915 


0.00 000 


57 


4 


7.06 579 


7.06 579 


2.93 421 


0.00 000 


56 


5 


7.16 270 


7.16 270 


2.83 730 


0.00 000 


55 


6 


7.24 188 


7.24 188 


2.75 812 


0.00 000 


54 


7 


7.30 882 


7.30 882 


2.69 118 


0.00 000 


53 


8 


7.36 682 


7.36 682 


2.63 318 


0.00 000 


52 


9 
10 

11 


7.41 797 


7.41 797 


2.58 203 


0.00 000 


51 
50 

49 


7.46 373 


7.46 373 


2.53 627 


0.00 000 


7.50 512 


7.50 512 


2.49 488 


0.00 000 


12 


7.54 291 


7.54 291 


2.45 709 


0.00 000 


48 


13 


7.57 767 


7.57 767 


2.42 233 


0.00 000 


47 


14 


7.60 985 


7.60 986 


2.39 014 


0.00 000 


46 


15 


7.63 982 


7.63 982 


2.36 018 


0.00 000 


45 


16 


7.66 784 


7.66 785 


2.33 215 


0.00 000 


44 


17 


7.69 417 


7.69 418 


2.30 582 


9.99 999 


43 


18 


7.71 900 


7.71 900 


2.28 100 


9.99 999 


42 


19 
20 

21 


7.74 248 


7.74 248 


2.25 752 


9.99 999 


41 
40 

39 


7.76 475 


7.76 476 


2.23 524 


9.99 999 


7.78 594 


7.78 595 


2.21 405 


9.99 999 


22 


7.80 615 


7.80 615 


2.19 385 


9.99 999 


38 


23 


7.82 545 


7.82 546 


2.17 454 


9.99 999 


37 


24 


7.84 393 


7.84 394 


2.15 606 


9.99 999 


36 


25 


7.86 166 


7.86 167 


2.13 833 


9.99 999 


35 


26 


7.87 870 


7.87 871 


2.12 129 


9.99 999 


34 


27 


7.89 509 


7.89 510 


2.10 490 


9.99 999 


33 


28 


7.91 088 


7.91 089 


2.08 911 


9.99 999 


32 


29 
30 

31 


7.92 612 


7.92 613 


2.07 387 


9.99 998 


31 
30 

29 


7.94 084 


7.94 086 


2.05 914 


9.99 998 


7.95 508 


7.95 510 


2.04 490 


9.99 998 


32 


7.96 887 


7.96 889 


2.03 111 


9.99 998 


28 


33 


7.98 223 


7.98 225 


2.01 775 


9.99 998 


27 


34 


7.99 520 


7.99 522 


2.00 478 


9.99 998 


26 


35 


8.00 779 


8.00 781 


1.99 219 


9.99 998 


25 


36 


8.02 002 


8.02 004 


1.97 996 


9.99 998 


24 


37 


8.03 192 


8.03 194 


1.96 806 


9.99 997 


23 


38 


8.04 350 


8.04 353 


1.95 647 


9.99 997 


22 


39 
40 
41 


8.05 478 


8.05 481 


1.94 519 


9.99 997 


21 
20 

19 


8.06 578 


8.06 581 


1.93 419 


9.99 997 


8.07 650 


8.07 653 


1.92 347 


9.99 997 


42 


8.08 696 


8.08 700 


1.91 300 


9.99 997 


18 


43 


8.09 718 


8.09 722 


1.90 278 


9.99 997 


17 


44 


8.10 717 


8.10 720 


1.89 280 


9.99 996 


16 


45 


8.11 693 


8.11 696 


1.88 304 


9.99 996 


15 


46 


8.12 647 


8.12 651 


1.87 349 


9.99 996 


14 


47 


8.13 581 


8.13 585 


1.86 415 


9.99 996 


13 


48 


8.14 495 


8.14 500 


1.85 500 


9.99 996 


12 


49 
50 

51 


8.15 391 


8.15 395 


1.84 605 


9.99 996 


11 
10 

9 


8.16 268 


8.16 273 


1.83 727 


9.99 995 


8.17 128 


8.17 133 


1.82 867 


9.99 995 


52 


8.17 971 


8.17 976 


1.82 024 


9.99 995 


8 


53 


8.18 798 


8.18 804 


1.81 196 


9.99 995 


7 


54 


8.19 610 


8.19 616 


1.80 384 


9.99 995 


6 


55 


8.20 407 


8.20 413 


1.79 587 


9.99 994 


5 


56 


8.21 189 


8.21 195 


1.78 805 


9.99 994 


4 


57 


8.21 958 


8.21 964 


1.78 036 


9.99 994 


3 


58 


8.22 713 


8.22 720 


1.77 280 


9.99 994 


2 


59 
60 


8.23 456 


8.23 462 


1.76 538 


9.99 994 


1 



8.24 186 


8.24 192 


1.75 808 


9.99 993 




L. Cos. 


L. Cot. 


L. Tan. 


L. Sin. 


t 






8! 


r 







28 







1 


o 






t 


L. Sin. 


L. Tan. 


L. Cot. 


L. Cos. 






1 


8.24 186 


8.24 192 


1.75 808 


9.99 993 


60 

59 


8.24 903 


8.24 910 


1.75 090 


9.99 993 


2 


8.25 609 


8.25 616 


1.74 384 


9.99 9D3 


58 


3 


8.26 304 


8.26 312 


1.73 688 


9.99 993 


57 


4 


8.26 988 


8.26 996 


1.73 004 


9.99 992 


56 


5 


8.27 661 


8.27 669 


1.72 331 


9.99 992 


55 


6 


8.28 324 


8.28 332 


1.71 668 


9.99 992 


54 


7 


8.28 977 


8.28 986 


1.71014 


9.99 992 


53 


8 


8.29 621 


8.29 629 


1.70 371 


9.99 992 


52 


9 
10 

11 


8.30 255 


8.30 263 


1.69 737 


9.99 991 


51 
50 

49 


8.30 879 


8.30 888 


1.69 112 


9.99 991 


8.31 495 


8.31 505 


1.68 495 


9.99 991 


12 


8.32 103 


8.32 112 


1.67 888 


9.99 990 


48 


13 


8.32 702 


8.32 711 


1.67 289 


9.99 990 


47 


14 


8.33 292 


8.33 302 


1.66 698 


9.99 990 


46 


15 


8.33 875 


8.33 886 


1.66 114 


9.99 990 


45 


16 


8.34 450 


8.34 461 


1.65 539 


9.99 989 


44 


17 


8.35 018 


8.35 029 


1.64 971 


9.99 989 


43 


18 


8.35 578 


8.35 590 


1.64 410 


9.99 989 


42 


19 
20 

21 


8.36 131 


8.36 143 


1.63 857 


9.99 989 


41 
40 

39 


8.36 678 


8.36 689 


1.63 311 


9.99 988 


8.37 217 


8.37 229 


1.62 771 


9.99 988 


22 


837 750 


8.37 762 


1.62 238 


9.99 988 


38 


23 


8.38 276 


8.38 289 


1.61 711 


9.99 987 


37 


24 


8.38 796 


8.38 809 


1.61 191 


9.99 987 


36 


25 


8.39 310 


8.39 323 


1.60 677 


9.99 987 


35 


26 


8.39 818 


8.39 832 


1.60 168 


9.99 986 


34 


27 


8.40 320 


8.40 334 


1.59 666 


9.99 986 


33 


28 


&.40 816 


8.40 830 


1.59 170 


9.99 986 


32 


29 
30 

31 


8.41 307 


8.41 321 


1.58 679 


9.99 985 


31 
30 

29 


8.41 792 


8.41 807 


1.58 193 


9.99 985 


8.42 272 


8.42 287 


1.57 713 


9.99 985 


32 


8.42 746 


8.42 762 


1.57 238 


9.99 984 


28 


33 


8.43 216 


8.43 232 


1.56 768 


9.99 984 


27 


34 


8.43 680 


8.43 696 


1.56 304 


9.99 984 


26 


35 


8.44 139 


8.44 156 


1.55 844 


9.99 983 


25 


36 


8.44 594 


8.44 611 


1.55 389 


9.99 983 


24 


37 


8.45 044 


8.45 061 


1.54 939 


9.99 983 


23 


38 


8.45 489 


8.45 507 


1.54 493 


9.99 982 


22 


39 
40 
41 


8.45 930 


8.45 948 


1.54 052 


9.99 982 


21 
20 

19 


8.46 366 


8.46 385 


1.53 615 


9.99 982 


8.46 799 


8.46 817 


1.53 183 


9.99 981 


42 


8.47 226 


8.47 245 


1.52 755 


9.99 981 


18* 


43 


8.47 650 


8.47 669 


1.52 331 


9.99 981 


17 


44 


8.48 069 


8.48 089 


1.51 911 


9.99 980 


16 


45 


8.48 485 


8.48 505 


1.51 495 


9.99 980 


15 


46 


8.48 896 


8.48 917 


1.51 083 


9.99 979 


14 


47 


8.49 304 


8.49 325 


1.50 675 


9.99 979 


13 


48 


8.49 708 


8.49 729 


1.50 271 


9.99 979 


12 


49 
50 
51 


8.50 108 


8.50 130 


1.49 870 


9.99 978 


11 
10 

9 


8.50 504 


8.50 527 


1.49 473 


9.99 978 


8.50 897 


8.50 920 


1.49 080 


9.99 977 


52 


8.51 287 


8.51 310 


1.48 690 


9.99 977 


8 


53 


8.51 673 


8.51 696 


1.48 304 


9.99 977 


7 


54 


8.52 055 


8.52 079 


1.47 921 


9.99 976 


6 


55 


8.52 434 


8.52 459 


1.47 541 


9.99 976 


5 


56 


8.52 810 


8.52 835 


1.47 165 


9.99 975 


4 


57 


8.53 183 


8.53 208 


1.46 792 


9.99 975 


3 


58 


8.53 552 


8.53 578 


1.46 422 


9.99 974 


2 


59 
60 


8.53 919 


8.53 945 


1.46 055 


9.99 974 


1 



8.54 282 


8.54 308 


1.45 692 


9.99 974 




L. Cos. 


L. Cot. 


L. Tan. 


L. Sin. 


t 



88' 







2 


o 






t 


L. Sin. 


L. Tan. 


L. Cot. 


L. Cos. 






1 


8.54 282 


8.54 308 


1.45 692 


9.99 974 


60 

59 


8.54 642 


8.54 669 


1.45 331 


9.99 973 


2 


8.54 999 


8.55 027 


1.44 973 


9.99 973 


58 


3 


8.55 354 


8.55 382 


1.44 618 


9.99 972 


57 


4 


S..55 705 


8.55 734 


1.44 266 


9.99 972 


56 


5 


8.56 054 


8.56 083 


1.43 917 


9.99 971 


55 


6 


8.56 400 


8.56 429 


1.43 571 


9.99 971 


54 


7 


8.56 743 


8.56 773 


1.43 227 


9.99 970 


53 


8 


8.57 084 


8.57 114 


1.42 886 


9.99 970 


52 


9 
10 

11 


8.57 421 


8.57 452 


1.42 548 


9.99 969 


51 
50 

49 


8.57 757 


8.57 788 


1.42 212 


9.99 969 


8.58 089 


8.58 121 


1.41 879 


9.99 968 


12 


8.58 419 


8.58 451 


1.41 549 


9.99 968 


48 


13 


8.58 747 


8.58 779 


1.41 221 


9.99 967 


47 


14 


8.59 072 


8.59 105 


1.40 895 


9.99 967 


46 


15 


8.59 395 


8.59 428 


1.40 572 


9.99 967 


45 


16 


8.59 715 


8.59 749 


1.40 251 


9.99 966 


44 


17 


8.60 033 


8.60 068 


1.39 932 


9.99 966 


43 


18 


8.60 349 


8.60 384 


1.39 616 


9.99 965 


42 


19 
20 

21 


8.60 662 


8.60 698 


1.39 302 


9.99 964 


41 
40 

39 


8.60 973 


8.61 009 


1.38 991 


9.99 964 


8.61 282 


8.61 319 


1.38 681 


9.99 963 


22 


8.61 589 


8.61 626 


1.38 374 


9.99 963 


38 


23 


8.61 894 


8.61 931 


1.38 069 


9.99 962 


37 


24 


8.62 196 


8.62 234 


1.37 766 


9.99 962 


36 


25 


8.62 497 


8.62 535 


1.37 465 


9.99 961 


35 


26 


8.62 795 


8.62 834 


1.37 166 


9.99 961 


34 


27 


8.63 091 


8.63 131 


1.36 869 


9.99 960 


33 


28 


8.63 385 


8.63 426 


1.36 574 


9.99 960 


32 


29 
30 

31 


8.63 678 


8.63 718 


1.36 282 


9.99 959 


31 
30 

29 


8.63 968 


8.64 009 


1.35 991 


9.99 959 


8.64 256 


8.64 298 


1.35 702 


9.99 958 


32 


8.64 543 


8.64 585 


1.35 415 


9.99 958 


28 


33 


8.64 827 


8.64 870 


1.35 130 


9.99 957 


27 


34 


8.65 110 


8.65 154 


1.34 846 


9.99 956 


26 


35 


8.65 391 


8.65 435 


1.34 565 


9.99 956 


25 


36 


8.65 670 


8.65 715 


1.34 285 


9.99 955 


24 


37 


8.65 947 


8.65 993 


1.34 007 


9.99 955 


23 


38 


8.66 223 


8.66 269 


1.33 731 


9.99 954 


22 


39 
40 
41 


8.66 497 


8.66 543 


1.33 457 


9.99 954 


21 
20 

19 


8.66 769 


8.66 816 


1.33 184 


9.99 953 


8.67 039 


8.67 087 


1.32 913 


9.99 952 


42 


8.67 308 


8.67 356 


1.32 644 


9.99 952 


18 


43 


8.67 575 


8.67 624 


1.32 376 


9.99 951 


17 


44 


8.67 841 


8.67 890 


1.32 110 


9.99 951 


16 


45 


8.68 104 


8.68 154 


1.31 846 


9.99 950 


15 


46 


8.68 367 


8.68 417 


1.31 583 


9.99 949 


14 


47 


8.68 627 


8.68 678 


1.31 322 


9.99 949 


13 


48 


8.68 886 


8.68 938 


1.31 062 


9.99 948 


12 


49 
50 

51 


8.69 144 


8.69 196 


1.30 804 


9.99 948 


11 
10 

9 


8.69 400 


8.69 453 


1.30 547 


9.99 947 


8.69 654 


8.69 708 


1.30 292 


9.99 946 


52 


8.69 907 


8.69 962 


1.30 038 


9.99 946 


8 


53 


8.70 159 


8.70 214 


1.29 786 


9.99 945 


7 


54 


8.70 409 


8.70 465 


1.29 535 


9.99 944 


6 


55 


8.70 658 


8.70 714 


1.29 286 


9.99 944 


5 


56 


8.70 905 


8.70 962 


1.29 038 


9.99 943 


4 


57 


8.71 151 


8.71 208 


1.28 792 


9.99 942 


3 


58 


8.71 395 


8.71 453 


1.28 547 


9.99 942 


2 


59 
60 


8.71 638 


8.71 697 


1.28 303 


9.99 941 


1 



8.71 880 


8.71 940 


1.28 060 


9.99 940 




L. Cos. 


L. Cot. 


L. Tan. 


L. Sin. 


/ 



29 



87' 



3° 



t 


L. Sin. 


L. Tan. 


L. Cot. 


L. Cos. 






1 


8.71 880 


8.71 940 


1.28 060 


9.99 940 


60 

59 


8.72 120 


8.72 181 


1.27 819 


9.99 940 


2 


8.72 359 


8.72 420 


1.27 580 


9.99 939 


58 


3 


8.72 597 


8.72 659 


1.27 341 


9.99 938 


57 


A 


8.72 834 


8.72 896 


1.27 104 


9.99 938 


56 


5 


8.73 069 


8.73 132 


1.26 868 


9.99 937 


55 


6 


8.73 303 


8.73 366 


1.26 634 


9.99 936 


54 


7 


8.73 535 


8.73 600 


1.26 400 


9.99 936 


53 


8 


8.73 767 


8.73 832 


1.26 168 


9.99 935 


52 


9 
10 
11 


8.73 997 


8.74 063 


1.25 937 


9.99 934 


51 
50 

49 


8.74 226 


8.74 292 


1.25 708 


9.99 934 


8.74 454 


8.74 521 


1.25 479 


9.99 933 


12 


8.74 680 


8.74 748 


1.25 252 


9.99 932 


48 


13 


8.74 906 


8.74 974 


1.25 026 


9.99 932 


47 


14 


8.75 130 


8.75 199 


1.24 801 


9.99 931 


46 


15 


8.75 353 


8.75 423 


1.24 577 


9.99 930 


45 


16 


8.75 575 


8.75 645 


1.24 355 


9.99 929 


44 


17 


8.75 795 


8.75 867 


1.24 133 


9.99 929 


43 


18 


8.76 015 


8.76 087 


1.23 913 


9.99 928 


42 


19 
20 
21 


8.76 234 


8.76 306 


1.23 694 


9.99 927 


41 
40 

39 


8.76 451 


8.76 525 


1.23 475 


9.99 926 


8.76 667 


8.76 742 


1.23 258 


9.99 926 


22 


8.76 883 


8.76 958 


1.23 042 


9.99 925 


38 


23 


8.77 097 


8.77 173 


1.22 827 


9.99 924 


37 


24 


8.77 310 


8.77 387 


1.22 613 


9.99 923 


36 


25 


8.77 522 


8.77 600 


1.22 400 


9.99 923 


35 


26 


8.77 733 


8.77 811 


1.22 189 


9.99 922 


34 


27 


8.77 943 


8.78 022 


1.21 978 


9.99 921 


33 


28 


8.78 152 


8.78 232 


1.21 768 


9.99 920 


32 


29 
30 

31 


8.78 360 


8.78 441 


1.21 559 


9.99 920 


31 
30 

29 


8.78 568 


8.78 649 


1.21 351 


9.99 919 


8.78 774 


8.78 855 


1.21 145 


9.99 918 


32 


8.78 979 


8.79 061 


1.20 939 


9.99 917 


28 


33 


8.79 183 


8.79 266 


1.20 734 


9.99 917 


27 


34 


8.79 386 


8.79 470 


1.20 530 


9.99 916 


26 


35 


8.79 588 


8.79 673 


1.20 327 


9.99 915 


25 


36 


8.79 789 


8.79 875 


1.20 125 


9.99 914 


24 


37 


8.79 990 


8.80 076 


1.19 924 


9.99 913 


23 


38 


8.80 189 


8.80 277 


, 1.19 723 


9.99 913 


22 


39 
40 

41 


8.80 388 


8.80 476 


1.19 524 


9.99 912 


21 
20 
19 


8.80 585 


8.80 674 


1.19 326 


9.99 911 


8.80 782 


8.80 872 


1.19 128 


9.99 910 


42 


8.80 978 


8.81 068 


1.18 932 


9.99 909 


18 


43 


8.81 173 


8.81 264 


1.18 736 


9.99 909 


17 


44 


8.81 367 


8.81 459 


1.18 541 


9.99 908 


16 


45 


8.81 560 


8.81 653 


1.18 347 


9.99 907 


15 


46 


8.81 752 


8.81 846 


1.18 154 


9.99 906 


14 


47 


8.81 944 


8.82 038 


1.17 962 


9.99 905 


13 


48 


8.82 134 


8.82 230 


1.17 770 


9.99 904 


12 


49 
50 

51 


8.82 324 


8.82 420 


1.17 580 


9.99 904 


11 
10 

9 


8.82 513 


8.82 610 


1.17 390 


9.99 903 


8.82 701 


8.82 799 


1.17 201 


9.99 902 


52 


8.82 888 


8.82 987 


1.17 013 


9.99 901 


8 


53 


8.83 075 


8.83 175 


1.16 825 


9.99 900 


7 


54 


8.83 261 


8.83 361 


1.16 639 


9.99 899 


6 


55 


8.83 446 


8.83 547 


1.16 453 


9.99 898 


5 


56 


8.83 630 


8.83 732 


1.16 268 


9.99 898 


4 


57 


8.83 813 


8.83 916 


1.16 084 


9.99 897 


3 


58 


8.83 996 


8.84 100 


1.15 900 


9.99 896 


2 


59 
60 


8.84 177 


8.84 282 


1.15 718 


9.99 895 


1 



8.84 358 


8.84 464 


1.15 536 


9.99 894 




L. Cos. 


L. Cot. 


L. Tan. 


L. Sin. 


t 



86 



3i 



t 


L. Sin. 


L. Tan. 


L. Cot. 


L. Cos. 






1 


8.84 358 


8.84 464 


1.15 536 


9.99 894 


60 

59 


8.84 539 


8.84 646 


1.15 354 


9.99 893 


2 


8.84 718 


8.84 826 


1.15 174 


9.99 892 


58 


3 


8.84 897 


8.85 006 


1.14 994 


9.99 891 


57 


4 


8.85 075 


8.85 185 


1.14 815 


9.99 891 


56 


5 


8.85 252 


8.85 363 


1.14 637 


9.99 890 


55 


6 


8.85 429 


8.85 540 


1.14 460 


9.99 889 


54 


7 


8.85 605 


8.85 717 


1.14 283 


9.99 888 


53 


8 


8.85 780 


8.85 893 


1.14 107 


9.99 887 


52 


9 
10 
11 


8.85 955 


8.86 069 


1.13 931 


9.99 886 


51 
50 

49 


8.86 128 


8.86 243 


1.13 757 


9.99 885 


8.86 301 


8.86 417 


1.13 583 


9.99 884 


12 


8.86 474 


8.86 591 


1.13 409 


9.99 883 


48 


13 


8.86 645 


8.86 763 


1.13 237 


9.99 882 


47 


14 


8.86 816 


8.86 935 


1.13 065 


9.99 881 


46 


15 


8.86 987 


8.87 106 


1.12 894 


9.99 880 


45 


16 


8.87 156 


8.87 277 


1.12 723 


9.99 879 


44 


17 


8.87 325 


8.87 447 


1.12 553 


9.99 879 


43 


18 


8.87 494 


8.87 616 


1.12 384 


9.99 878 


42 


19 
20 
21 


8.87 661 


8.87 785 


1.12 215 


9.99 877 


41 
40 

39 


8.87 829 


8.87 953 


1.12 047 


9.99 876 


8.87 995 


8.88 120 


1.11 880 


9.99 875 


22 


8.88 161 


8.88 287 


1.11 713 


9.99 874 


38 


23 


8.88 326 


8.88 453 


1.11 547 


9.99 873 


37 


24 


8.88 490 


8.88 618 


1.11 382 


9.99 872 


36 


25 


8.88 654 


8.88 783 


1.11 217 


9.99 871 


35 


26 


8.88 817 


8.88 948 


1.11 052 


9.99 870 


34 


27 


8.88 980 


8.89 111 


1.10 889 


9.99 869 


33 


28 


8.89 142 


8.89 274 


1.10 726 


9.99 868 


32 


29 
30 

31 


8.89 304 


8.89 437 


1.10 563 


9.99 867 


31 
30 

29 


8.89 464 


8.89 598 


1.10 402 


9.99 866 


8.89 625 


8.89 760 


1.10 240 


9.99 865 


32 


8.89 784 


8.89 920 


1.10 080 


9.99 864 


28 


33 


8.89 943 


8.90 080 


1.09 920 


9.99 863 


27 


34 


8.90 102 


8.90 240 


1.09 760 


9.99 862 


26 


35 


8.90 260 


8.90 399 


1.09 601 


9.99 861 


25 


36 


8.90 417 


8.90 557 


1.09 443 


9.99 860 


24 


37 


8.90 574 


8.90 715 


1.09 285 


9.99 859 


23 


38 


8.90 730 


8.90 872 


1.09 128 


9.99 858 


22 


39 
40 

41 


8.90 885 


8.91 029 


1.08 971 


9.99 857 


21 
20 

19 


8.91 040 


8.91 185 


1.08 815 


9.99 856 


8.91 195 


8.91 340 


1.08 660 


9.99 855 


42 


8.91 349 


8.91 495 


1.08 505 


9.99 854 


18 


43 


8.91 502 


8.91 650 


1.08 350 


9.99 853 


17 


44 


8.91 655 


8.91 803 


1.08 197 


9.99 852 


16 


45 


8.91 807 


8.91 957 


1.08 043 


9.99 851 


15 


46 


8.91 959 


8.92 110 


1.07 890 


9.99 850 


14 


47 


8.92 110 


8.92 262 


1.07 738 


9.99 848 


13 


48 


8.92 261 


8.92 414 


1.07 586 


9.99 847 


12 


49 
50 

51 


8.92 411 


8.92 565 


1.07 435 


9.99 846 


11 
10 

9 


8.92 561 


8.92 716 


1.07 284 


9.99 845 


8.92 710 


8.92 866 


1.07 134 


9.99 844 


52 


8.92 859 


8.93 016 


1.06 984 


9.99 843 


8 


53 


8.93 007 


8.93 165 


1.06 835 


9.99 842 


7 


54 


8.93 154 


8.93 313 


1.06 687 


9.99 841 


6 


55 


8.93 301 


8.93 462 


1.06 538 


9.99 840 


5 


56 


8.93 448 


8.93 609 


1.06 391 


9.99 839 


4 


57 


8.93 594 


8.93 756 


1.06 244 


9.99 838 


3 


58 


8.93 740 


8.93 903 


1.06 097 


9.99 837 


2 


59 
60 


8.93 885 


8.94 049 


1.05 951 


9.99 836 


1 



8.94 030 


8.94 195 


1.05 805 


9.99 834 




L. Cos. 


L. Cot. 


L. Tan. 


L. Sin. 


/ 



85 



3 2 



t 


L. Sin. 


L. Tan. 


L. Cot. 


L. Cos. 






1 


8.94 030 


8.94 195 


1.05 805 


9.99 834 


60 

59 


8.94 174 


8.94 340 


1.05 660 


9.99 833 


2 


8.94 317 


8.94 485 


1.05 515 


9.99 832 


58 


3 


8.94 461 


8.94 630 


1.05 370 


9.99 831 


57 


4 


8.94 603 


8.94 773 


1.05 227 


9.99 830 


56 


5 


8.94 746 


8.94 917 


1.05 083 


9.99 829 


55 


6 


8.94 887 


8.95 060 


1.04 940 


9.99 828 


54 


7 


8.95 029 


8.95 202 


1.04 798 


9.99 827 


53 


8 


8.95 170 


8.95 344 


1.04 656 


9.99 825 


52 


9 
10 

11 


8.95 310 


8.95 486 


1.04 514 


9.99 824 


51 
50 

49 


8.95 450 


8.95 627 


1.04 373 


9.99 823 


8.95 589 


8.95 767 


1.04 233 


9.99 822 


12 


8.95 728 


8.95 908 


1.04 092 


9.99 821 


48 


13 


8.95 867 


8.96 047 


1.03 953 


9.99 820 


47 


14 


8.96 005 


8.96 187 


1.03 813 


9.99 819 


46 


15 


8.96 143 


8.96 325 


1.03 675 ' 


9.99 817 


45 


16 


8.96 280 


8.96 464 


1.03 536 


9.99 816 


44 


17 


8.96 417 


8.96 602 


1.03 398 


9.99 815 


43 


18 


8.96 553 


8.96 739 


1.03 261 


9.99 814 


42 


19 
20 
21 


8.96 689 


8.96 877 


1.03 123 


9.99 813 


41 
40 

39 


8.96 825 


8.97 013 


1.02 987 


9.99 812 


8.96 960 


8.97 150 


1.02 850 


9.99 810 


22 


8.97 095 


8.97 285 


1.02 715 


9.99 809 


38 


23 


8.97 229 


8.97 421 


1.02 579 


9.99 808 


37 


24 


8.97 363 


8.97 556 


1.02 444 


9.99 807 


36 


25 


8.97 496 


8.97 691 


1.02 309 


9.99 806 


35 


26 


8.97 629 


8.97 825 


1.02 175 


9.99 804 


34 


27 


8.97 762 


8.97 959 


1.02 041 


9.99 803 


33 


28 


8.97 894 


8.98 092 


1.01 908 


9.99 802 


32 


29 
30 

31 


8.98 026 


8.98 225 


1.01 775 


9.99 801 


31 
30 

29 


8.98 157 


8.98 358 


1.01 642 


9.99 800 


8.98 288 


8.98 490 


1.01 510 


9.99 798 


32 


8.98 419 


8.98 622 


1.01 378 


9.99 797 


28 


33 


8.98 549 


8.98 753 


1.01 247 


9.99 796 


27 


34 


8.98 679 


8.98 884 


1.01 116 


9.99 795 


26 


35 


8.98 808 


8.99 015 


1.00 985 


9.99 793 


25 


36 


8.98 937 


8.99 145 


1.00 855 


9.99 792 


24 


37 


8.99 066 


8.99 275 


1.00 725 


9.99 791 


23 


38 


8.99 194 


8.99 405 


1.00 595 


9.99 790 


22 


39 
40 

41 


8.99 322 


8.99 534 


1.00 466 


9.99 788 


21 
20 

19 


8.99 4o0 


8.99 662 


1.00 338 


9.99 787 


8.99 577 


8.99 791 


1.00 209 


9.99 786 


42 


8.99 704 


8.99 919 


1.00 081 


9.99 785 


18 


43 


8.99 830 


9.00 046 


0.99 954 


9.99 783 


17 


44 


8.99 956 


9.00 174 


0.99 826 


9.99 782 


16 


45 


9.00 082 


9.00 301 


0.99 699 


9.99 781 


15 


46 


9.00 207 


9.00 427 


0.99 573 


9.99 780 


14 


47 


9.00 332 


9.00 553 


0.99 447 


9.99 778 


13 


48 


9.00 456 


9.00 679 


0.99 321 


9.99 777 


12 


49 
50 

51 


9.00 581 


9.00 805 


0.99 195 


9.99 776 


11 
10 

9 


9.00 704 


9.00 930 


0.99 070 


9.99 775 


9.00 828 


9.01 055 


0.98 945 


9.99 773 


52 


9.00 951 


9.01 179 


0.98 821 


9.99 772 


8 


53 


9.01 074 


9.01 303 


0.98 697 


9.99 771 


7 


54 


9.01 196 


9.01 427 


0.98 573 


9.99 769 


6 


55 


9.01 318 


9.01 550 


0.98 450 


9.99 768 


5 


56 


9.01 440 


9.01 673 


0.98 327 


9.99 767 


4 


57 


9.01 561 


9.01 796 


0.98 204 


9.99 765 


3 


58 


9.01 682 


9.01 918 


0.98 082 


9.99 764 


2 


59 
60 


9.01 803 


9.02 040 


0.97 960 


9.99 763 


1 



9.01923 


9.02 162 


0.97 838 


9.99 761 




L. Cos. 


L, Cot. 


L. Tan. 


L. Sin. 


t 



84' 







6° 






1 


L. Sin. 


L. Tan. 


L. Cot. 


L. Cos. 






1 


9.01923 


9.02 162 


0.97 838 


9.99 761 


60 

59 


9.02 043 


9.02 283 


0.97 717 


9.99 760 


2 


9.02 163 


9.02 404 


0.97 596 


9.99 759 


58 


3 


9.02 283 


9.02 525 


0.97 475 


9.99 757 


57 


4 


9.02 402 


9.02 645 


0.97 355 


9.99 756 


56 


5 


9.02 520 


9.02 766 


0.97 234 


9.99 755 


55 


6 


9.02 639 


9.02 885 


0.97 115 


9.99 753 


54 


7 


9.02 757 


9.03 005 


0.96 995 


9.99 752 


53 


8 


9.02 874 


9.03 124 


0.96 876 


9.99 751 


52 


9 
10 
11 


9.02 992 


9.03 242 


0.96 758 


9.99 749 


51 
50 

49 


9.03 109 


9.03 361 


0.96 639 


9.99 748 


9.03 226 


9.03 479 


0.96 521 


9.99 747 


12 


9.03 342 


9.03 597 


0.96 403 


9.99 745 


48 


13 


9.03 458 


9.03 714 


0.96 286 


9.99 744 


47 


14 


9.03 574 


9.03 832 


0.96 168 


9.99 742 


46 


15 


9.03 690 


9.03 948 


0.96 052 


9.99 741 


45 


16 


9.03 805 


9.04 065 


0.95 935 


9.99 740 


44 


17 


9.03 920 


9.04 181 


0.95 819 


9.99 738 


43 


18 


9.04 034 


9.04 297 


0.95 703 


9.99 737 


42 


19 
20 

21 


9.04 149 


9.04 413 


0.95 587 


9.99 736 


41 
40 

39 


9.04 262 


9.04 528 


0.95 472 


9.99 734 


9.04 376 


9.04 643 


0.95 357 


9.99 733 


22 


9.04 490 


9.04 758 


0.95 242 


9.99 731 


38 


23 


9.04 603 


9.04 873 


0.95 127 


9.99 730 


37 


24 


9.04 715 


9.04 987 


0.95 013 


9.99 728 


36 


25 


9.04 828 


9.05 101 


0.94 899 


9.99 727 


35 


26 


9.04 940 


9.05 214 


0.94 786 


9.99 726 


34 


27 


9.05 052 


9.05 328 


0.94 672 


9.99 724 


33 


28 


9.05 164 


9.05 441 


0.94 559 


9.99 723 


32 


29 
30 

31 


9.05 275 


9.05 553 


0.94 447 


9.99 721 


31 
30 

29 


9.05 386 


9.05 666 


0.94 334 


9.99 720 


9.05 497 


9.05 778 


0.94 222 


9.99 718 


32 


9.05 607 


9.05 890 


0.94 110 


9.99 717 


28 


33 


9.05 717 


9.06 002 


0.93 998 


9.99 716 


27 


34 


9.05 827 


9.06 113 


0.93 887 


9.99 714 


26 


35 


9.05 937 


9.06 224 


0.93 776 


9.99 713 


25 


36 


9.06 046 


9.06 335 


0.93 665 


9.99 711 


24 


37 


9.06 155 


9.06 445 


0.93 555 


9.99 710 


23 


38 


9.06 264 


9.06 556 


0.93 444 


9.99 708 


22 


39 
40 

41 


9.06 372 


9.06 666 


0.93 334 


9.99 707 


21 
20 

19 


9.06 481 


9.06 775 


0.93 225 


9.99 705 


9.06 589 


9.06 885 


0.93 115 


9.99 704 


42 


9.06 696 


9.06 994 


0.93 006 


9.99 702 


18 


43 


9.06 804 


9.07 103 


0.92 897 


9.99 701 


17 


44 


9.06 911 


9.07 211 


0.92 789 


9.99 699 


16 


45 


9.07 018 


9.07 320 


0.92 680 


9.99 698 


15 


46 


9.07 124 


9.07 428 


0.92 572 


9.99 696 


14 


47 


9.07 231 


9.07 536 


0.92 464 


9.99 695 


13 


48 


9.07 337 


9.07 643 


0.92 357 


9.99 693 


12 


49 
50 

51 


9.07 442 


9.07 751 


0.92 249 


9.99 692 


11 
10 

9 


9.07 548 


9.07 858 


0.92 142 


9.99 690 


9.07 653 


9.07 964 


0.92 036 


9.99.689 


52 


9.07 758 


9.08 071 


0.91 929 


9.99 687 


8 


53 


9.07 863 


9.08 177 


0.91 823 


9.99 686 


7 


54 


9.07 968 


9.08 283 


0.91 717 


9.99 684 


6 


55 


9.08 072 


9.08 389 


0.91 611 


9.99 683 


5 


56 


9.08 176 


9.08 495 


0.91 505 


9.99 681 


4 


57 


9.08 280 


9.08 600 


0.91 400 


9.99 680 


3 


58 


9.08 383 


9.08 705 


0.91 295 


9.99 678 


2 


59 
60 


9.08 486 


9.08 810 


0.91 190 


9.99 677 


1 



9.08 589 


9.08 914 


0.91 086 


9.99 675 




L. Cos. 


L. Cot. 


L. Tan. 


L. Sin. 


/ 



33 



83' 



34 



t 


L. Sin. 


L. Tan. 


L. Cot. 


L. Cos. 






1 


9.08 589 


9.08 914 


0.91 086 


9.99 675 


60 

59 


9.08 692 


9.09 019 


0.90 981 


9.99 674 


2 


9.08 795 


9.09 123 


0.90 877 


9.99 672 


58 


3 


9.08 897 


9.09 227 


0.90 773 


9.99 670 


57 


4 


9.08 999 


9.09 330 


0.90 670 


9.99 669 


56 


5 


9.09 101 


9.09 434 


0.90 566 


9.99 667 


55 


6 


9.09 202 


9.09 537 


0.90 463 


9.99 666 


54 


7 


9.09 304 


9.09 640 


0.90 360 


9.99 664 


53 


8 


9.09 405 


9.09 742 


0.90 258 


9.99 663 


52 


9 
10 

11 


9.09 506 


9.09 845 


0.90 155 


9.99 661 


51 
50 

49 


9.09 606 


9.09 947 


0.90 053 


9.99 659 


9.09 707 


9.10 049 


0.89 951 


9.99 658 


12 


9.09 807 


9.10 150 


0.89 850 


9.99 656 


48 


13 


9.09 907 


9.10 252 


0.89 748 


9.99 655 


47 


14 


9.10 006 


9.10 353 


0.89 647 


9.99 653 


46 


15 


9.10 106 


9.10 454 


0.89 546 


9.99 651 


45 


16 


9.10 205 


9.10 555 


0.89 445 


9.99 650 


44 


17 


9.10 304 


9.10 656 


0.98 344 


9.99 648 


43 


18 


9.10 402 


9.10 756 


0.89 244 


9.99 647 


42 


19 
20 

21 


9.10 501 


9.10 856 


0.89 144 


9.99 645 


41 
40 

39 


9.10 599 


9.10 956 


0.89 044 


9.99 643 


9.10 697 


9.11 056 


0.88 944 


9.99 642 


22 


9.10 795 


9.11 155 


0.88 845 


9.99 640 


38 


23 


9.10 893 


9.11 254 


0.88 746 


9.99 638 


37 


24 


9.10 990 


9.11 353 


0.88 647 


9.99 637 


36 


25 


9.11 087 


9.11 452 


0.88 548 


9.99 635 


35 


26 


9.11 184 


9.11 551 


0.88 449 


9.99 633 


34 


27 


9.11 281 


9.11 649 


0.88 351 


9.99 632 


33 


28 


9.11 377 


9.11 747 


0.88 253 


1 9.99 630 


32 


29 
30 
31 


9.11 474 


9.11 845 


0.88 155 


9.99 629 


31 
30 

29 


9.11 570 


9.11 943 


0.88 057 


9.99 627 


9.11 666 


9.12 040 


0.87 960 


9.99 625 


32 


9.11 761 


9.12 138 


0.87 862 


9.99 624 


28 


33 


9.11 857 


9.12 235 


0.87 765 


9.99 622 


27 


34 


9.11 952 


9.12 332 


0.87 668 


9.99 620 


26 


35 


9.12 047 


9.12 428 


0.87 572 


9.99 618 


25 


36 


9.12 142 


9.12 525 


0.87 475 


9.99 617 


24 


37 


9.12 236 


9.12 621 


0.87 379 


9.99 615 


23 


38 


9.12 331 


9.12 717 


0.87 283 


9.99 613 


22 


39 
40 

41 


9.12 425 


9.12 813 


0.87 187 


9.99 612 


21 
20 

19 


9.12 519 


9.12 909 


0.87 091 


9.99 610 


9.12 612 


9.13 004 


0.86 996 


9.99 608 


42 


9.12 706 


9.13 099 


0.86 901 


9.99 607 


18 


43 


9.12 799 


9.13 194 


0.86 806 


9.99 605 


17 


44 


9.12 892 


9.13 289 


0.86 711 


9.99 603 


16 


45 


9.12 985 


9.13 384 


0.86 616 


9.99 601 


15 


46 


9.13 078 


9.13 478 


0.86 522 


9.99 600 


14 


47 


9.13 171 


9.13 573 


0.86 427 


9.99 598 


13 


48 


9.13 263 


9.13 667 


0.86 333 


9.99 596 


12 


49 
50 
51 


9.13 355 


9.13 761 


0.86 239 


9.99 595 


11 
10 

9 


9.13 447 


9.13 854 


0.86 146 


9.99 593 


9.13 539 


9.13 948 


0.86 052 


9.99 591 


52 


9.13 630 


9.14 041 


0.85 959 


9.99 589 


8 


53 


9.13 722 


9.14 134 


0.85 866 


9.99 588 


7 


54 


9.13 813 


9.14 227 


0.85 773 


9.99 586 


6 


55 


9.13 904 


9.14 320 


0.85 680 


9.99 584 


5 


56 


9.13 994 


9.14 412 


0.85 588 


9.99 582 


4 


57 


9.14 085 


9.14 504 


0.85 496 


9.99 581 


3 


58 


9.14 175 


9.14 597 


0.85 403 


9.99 579 


2 


59 
60 


9.14 266 


9.14 688 


0.85 312 


9.99 577 


1 



9.14 356 


9.14 780 


0.85 220 


9.99 575 




L. Cos. 


L. Cot. 


L. Tan. 


L. Sin. 


r 



82 



8' 



35 



t 


L. Sin. 


L. Tan. 


L. Cot. 


L. Cos. 






1 


9.14 356 


9.14 780 


0.85 220 


9.99 575 


60 

59 


9.14 445 


9.14 872 


0.85 128 


9.99 574 


2 


9.14 535 


9.14 963 


0.85 037 


9.99 572 


58 


3 


9.14 624 


9.15 054 


0.84 946 


9.99 570 


57 


4 


9.14 714 


9.15 145 


0.84 855 


9.99 568 


56 


5 


9.14 803 


9.15 236 


0.84 764 


9.99 566 


55 


6 


9.14 891 


9.15 327 


0.84 673 


9.99 565 


54 


7 


9.14 980 


9.15 417 


0.84 583 


9.99 563 


53 


8 


9.15 069 


9.15 508 


0.84 492 


9.99 561 


52 


9 
10 

11 


9.15 157 


9.15 598 


0.84 402 


9.99 559 


51 
50 

49 


9.15 245 


9.15 688 


0.84 312 


9.99 557 


9.15 333 


9.15 777 


0.84 223 


9.99 556 


12 


9.15 421 


9.15 867 


0.84 133 


9.99 554 


48 


13 


9.15 508 


9.15 956 


0.84 044 


9.99 552 


47 


14 


9.15 596 


9.16 046 


0.83 954 


9.99 550 


46 


15 


9.15 683 


9.16 135 


0.83 865 


9.99 548 


45 


16 


9.15 770 


9.16 224 


0.83 776 


9.99 546 


44 


17 


9.15 857 


9.16 312 


0.83 688 


9.99 545 


43 


18 


9.15 944 


9.16 401 


0.83 599 


9.99 543 


42 


19 
20 
21 


9.16 030 


9.16 489 


0.83 511 


9.99 541 


41 
40 

39 


9.16 116 


9.16 577 


0.83 423 


9.99 539 


9.16 203 


9.16 665 


0.83 335 


9.99 537 


22 


9.16 289 


9.16 753 


0.83 247 


9.99 535 


38 


23 


9.16 374 


9.16 841 


0.83 159 


9.99 533 


37 


24 


9.16 460 


9.16 928 


0.83 072 


9.99 532 


36 


25 


9.16 545 


9.17 016 


0.82 984 


9.99 530 


35 


26 


9.16 631 


9.17 103 


0.82 897 


9.99 528 


34 


27 


9.16 716 


9.17 190 


0.82 810 


9.99 526 


33 


28 


9.16 801 


9.17 277 


0.82 723 


9.99 524 


32 


29 
30 

31 


9.16 886 


9.17 363 


0.82 637 


9.99 522 


31 
30 

29 


9.16 970 


9.17 450 


0.82 550 


9.99 520 


9.17 055 


9.17 536 


0.82 464 


9.99 518 


32 


9.17 139 


9.17 622 


0.82 378 


9.99 517 


28 


33 


9.17 223 


9.17 708 


0.82 292 


9.99 515 


27 


34 


9.17 307 


9.17 794 


0.82 206 


9.99 513 


26 


35 


9.17 391 


9.17 880 


0.82 120 


9.99 511 


25 


36 


9.17 474 


9.17 965 


0.82 035 


9.99 509 


24 


37 


9.17 558 


9.18 051 


0.81 949 


9.99 507 


23 


38 


9.17 641 


9.18 136 


0.81 864 


9.99 505 


22 


39 
40 

41 


9.17 724 


9.18 221 


0.81 779 


9.99 503 


21 
20 

19 


9.17 807 


9.18 306 


0.81 694 


9.99 501 


9.17 890 


9.18 391 


0.81 609 


9.99 499 


42 


9.17 973 


9.18 475 


0.81 525 


9.99 497 


18 


43 


9.18 055 


9.18 560 


0.81 440 


9.99 495 


17 


44 


9.18 137 


9.18 644 


0.81 356 


9.99 494 


16 


45 


9.18 220 


9.18 728 


0.81 272 


9.99 492 


15 


46 


9.18 302 


9.18 812 


0.81 188 


9.99 490 


14 


47 


9.18 383 


9.18 896 


0.81 104 


9.99 488 


13 


48 


9.18 465 


9.18 979 


0.81 021 


9.99 486 


12 


49 
50 

51 


9.18 547 


9.19 063 


0.80 937 


9.99 484 


11 
10 

9 


9.18 628 


9.19 146 


0.80 854 


9.99 482 


9.18 709 


9.19 229 


0.80 771 


9.99 480 


52 


9.18 790 


9.19 312 


0.80 688 


9.99 478 


8 


53 


9.18 871 


9.19 395 


0.80 605 


9.99 476 


7 


54 


9.18 952 


9.19 478 


0.80 522 


9.99 474 


6 


55 


9.19 033 


9.19 561 


0.80 439 


9.99 472 


5 


56 


9.19 113 


9.19 643 


0.80 357 


9.99 470 


4 


57 


9.19 193 


9.19 725 


0.80 275 


9.99 468 


3 


58 


9.19 273 


9.19 807 


0.80 193 


9.99 466 


2 


59 
60 


9.19 353 


9.19 889 


0.80 111 


9.99 464 


1 



9.19 433 


9.19 971 


0.80 029 


9.99 462 




L. Cos. 


L. Cot. 


L. Tan. 


L. Sin. 


t 



81 c 



36 



/ 


L. Sin. 


L. Tan. 


L. Cot. 


L. Cos. 






1 


9.19 433 


9.19 971 


0.80 029 


9.99 462 


60 

59 


9.19 513 


9.20 053 


0.79 947 


9.99 460 


2 


9.19 592 


9.20 134 


0.79 866 


9.99 458 


58 


3 


9.19 672 


9.20 216 


0.79 784 


9.99 456 


57 


4 


9.19 751 


9.20 297 


0.79 703 


9.99 454 


56 


5 


9.19 830 


9.20 378 


0.79 622 


9.99 452 


55 


6 


9.19 909 


9.20 459 


0.79 541 


9.99 450 


54 


7 


9.19 988 


9.20 540 


0.79 460 


9.99 448 


53 


8 


9.20 067 


9.20 621 


0.79 379 


9.99 446 


52 


9 
10 

11 


9.20 145 


9.20 701 


0.79 299 


9.99 444 


51 
50 

49 


9.20 223 


9.20 782 


0.79 218 


9.99 442 


9.20 302 


9.20 862 


0.79 138 


9.99 440 


12 


9.20 380 


9.20 942 


0.79 058 


9.99 438 


48 


13 


9.20 458 


9.21 022 


0.78 978 


9.99 436 


47 


14 


9.20 535 


9.21 102 


0.78 898 


9.99 434 


46 


15 


9.20 613 


9.21 182 


0.78 818 


9.99 432 


45 


16 


9.20 691 


9.21 261 


0.78 739 


9.99 429 


44 


17 


9.20 768 


9.21 341 


0.78 659 


9.99 427 


43 


18 


9.20 845 


9.21 420 


0.78 580 


9.99 425 


42 


19 
20 
21 


9.20 922 


9.21 499 


0.78 501 


9.99 423 


41 
40 
39 


9.20 999 


9.21 578 


0.78 422 


9.99 421 


9.21 076 


9.21 657 


0.78 343 


9.99 419 


22 


9.21 153 


9.21 736 


0.78 264 


9.99 417 


38 


23 


9.21 229 


9.21 814 


0.78 186 


9.99 415 


37 


24 


9.21 306 


9.21 893 


0.78 107 


9.99 413 


36 


25 


9.21 382 


9.21 971 


0.78 029 


9.99 411 


35 


26 


9.21 458 


9.22 049 


0.77 951 


9.99 409 


34 


27 


9.21 534 


9.22 127 


0.77 873 


9.99 407 


33 


28 


9.21 610 


9.22 205 


0.77 795 


9.99 404 


32 


29 
30 

31 


9.21 685 


9.22 283 


0.77 717 


9.99 402 


31 
30 

29 


9.21 761 


9.22 361 


0.77 639 


9.99 400 


9.21 836 


9.22 438 


0.77 562 


9.99 398 


32 


9.21 912 


9.22 516 


0.77 484 


9.99 396 


28 


33 


9.21 987 


9.22 593 


0.77 407 


9.99 394 


27 


34 


9.22 062 


9.22 670 


0.77 330 


9.99 392 


26 


35 


9.22 137 


9.22 747 


0.77 253 


9.99 390 


25 


36 


9.22 211 


9.22 824 


0.77 176 


9.99 388 


24 


37 


9.22 286 


9.22 901 


0.77 099 


9.99 385 


23 


38 


9.22 361 


9.22 977 


0.77 023 


9.99 383 


22 


39 
40 
41 


9.22 435 


9.23 054 


0.76 946 


9.99 381 


21 
20 

19 


9.22 509 


9.23 130 


0.76 870 


9.99 379 


9.22 583 


9.23 206 


0.76 794 


9.99 377 


42 


9.22 657 


9.23 283 


0.76 717 


9.99 375 


18 


43 


9.22 731 


9.23 359 


0.76 641 


9.99 372 


17 


44 


9.22 805 


9.23 435 


0.76 565 


9.99 370 


16 


45 


9.22 878 


9.23 510 


0.76 490 


9.99 368 


15 


46 


9.22 952 


9.23 586 


0.76 414 


9.99 366 


14 


47 


9.23 025 


9.23 661 


0.76 339 


9.99 364 


13 


48 


9.23 098 


9.23 737 


0.76 263 


9.99 362 


12 


49 
50 
51 


9.23 171 


9.23 812 


0.76 188 


9.99 359 


11 
10 

9 


9.23 244 


9.23 887 


0.76 113 


9.99 357 


9.23 317 


9.23 962 


0.76 038 


9.99 355 


52 


9.23 390 


9.24 037 


0.75 963 


9.99 353 


8 


53 


9.23 462 


9.24 112 


0.75 888 


9.99 351 


7 


54 


9.23 535 


9.24 186 


0.75 814 


9.99 348 


6 


55 


9.23 607 


9.24 261 


0.75 739 


9.99 346 


5 


56 


9.23 679 


9.24 335 


0.75 665 


9.99 344 


4 


57 


9.23 752 


9.24 410 


0.75 590 


9.99 342 


3 


58 


9.23 823 


9.24 484 


0.75 516 


9.99 340 


2 


59 
60 


9.23 895 


9.24 558 


0.75 442 


9.99 337 


1 



9.23 967 


9.24 632 


0.75 368 


9.99 335 




L. Cos. 


L. Cot. 


L. Tan. 


L. Sin. 


/ 



80 



10 c 



37 



t 


L. Sin. 


L. Tan. 


L. Cot. 


L. Cos. 






1 


9.23 967 


9.24 632 


0.75 368 


9.99 335 


60 

59 


9.24 039 


9.24 706 


0.75 294 


9.99 333 


2 


9.24 110 


9.24 779 


0.75 221 


9.99 331 


58 


3 


9.24 181 


9.24 853 


0.75 147 


9.99 328 


57 


4 


9.24 253 


9.24 926 


0.75 074 


9.99 326 


56 


5 


9.24 324 


9.25 000 


0.75 000 


9.99 324 


55 


6 


9.24 395 


9.25 073 


0.74 927 


9.99 322 


54 


7 


9.24 466 


9.25 146 


0.74 854 


9.99 319 


53 


8 


9.24 536 


9.25 219 


0.74 781 


9.99 317 


52 


9 
10 

11 


9.24 607 


9.25 292 


0.74 708 


9.99 315 


51 
50 

49 


9.24 677 


9.25 365 


0.74 635 


9.99 313 


9.24 748 


9.25 437 


0.74 563 


9.99 310 


12 


9.24 818 


9.25 510 


0.74 490 


9.99 308 


48 


13 


9.24 888 


9.25 582 - 


0.74 418 


9.99 306 


47 


14 


9.24 958 


9.25 655 


0.74 345 


9.99 304 


46 


15 


9.25 028 


9.25 727 


0.74 273 


9.99 301 


45 


16 


9.25 098 


9.25 799 


0.74 201 


9.99 299 


44 


17 


9.25 168 


9.25 871 


0.74 129 


9.99 297 


43 


18 


9.25 237 


9.25 943 


0.74 057 


9.99 294 


42 


19 
20 
21 


9.25 307 


9.26 015 


0.73 985 


9.99 292 


41 
40 

39 


9.25 376 


9.26 086 


0.73 914 


9.99 290 


9.25 445 


9.26 158 


0.73 842 


9.99 288 


22 


9.25 514 


9.26 229 


0.73 771 


9.99 285 


38 


23 


9.25 583 


9.26 301 


0.73 699 


9.99 283' 


37 


24 


9.25 652 


9.26 372 


0.73 628 


9.99 281 


36 


25 


9.25 721 


9.26 443 


0.73 557 


9.99 278 


35 


26 


9.25 790 


9.26 514 


0.73 486 


9.99 276 


34 


27 


9.25 858 


9.26 585 


0.73 415 


9.99 274 


33 


28 


9.25 927 


9.26 655 


0.73 345 


9.99 271 


32 


29 
30 

31 


9.25 995 


9.26 726 


0.73 274 


9.99 269 


31 
30 

29 


9.26 063 


9.26 797 


0.73 203 


9.99 267 


9.26 131 


9.26 867 


0.73 133 


9.99 264 


32 


9.26 199 


9.26 937 


0.73 063 


9.99 262 


28 


33 


9.26 267 


9.27 008 


0.72 992 


9.99 260 


27 


34 


9.26 335 


9.27 078 


0.72 922 


9.99 257 


26 


35 


9.26 403 


9.27 148 


0.72 852 


. 9.99 255 


25 


36 


9.26 470 


9.27 218 


0.72 782 


9.99 252 


24 


37 


9.26 538 


9.27 288 


0.72 712 


9.99 250 


23 


38 


9.26 605 


9.27 357 


0.72 643 


9.99 248 


22 


39 
40 

41 


9.26 672 


9.27 427 


0.72 573 


9.99 245 


21 
20 

19 


9.26 739 


9.27 496 


0.72 504 


9.99 243 


9.26 806 


9.27 566 


0.72 434 


9.99 241 


42 


9.26 873 


9.27 635 


0.72 365 


9.99 238 


18 


43 


9.26 940 


9.27 704 


0.72 296 


9.99 236 


17 


44 


9.27 007 


9.27 773 


0.72 227 


9.99 233 


16 


45 


9.27 073 


9.27 842 


0.72 158 


9.99 231 


15 


46 


9.27 140 


9.27 911 


0.72 089 


9.99 229 


14 


47 


9.27 206 


9.27 980 


0.72 020 


9.99 226 


13 


48 


9.27 273 


9.28 049 


0.71 951 


9.99 224 


12 


49 
50 

51 


9.27 339 


9.28 117 


0.71 883 


9.99 221 


11 
10 

9 


9.27 405 


9.28 186 


0.71 814 


9.99 219 


9.27 471 


9.28 254 


0.71 746 


9.99 217 


52 


9.27 537 


9.28 323 


0.71 677 


9.99 214 


8 


53 


9.27 602 


9.28 391 


0.71 609 


9.99 212 


7 


54 


9.27 668 


9.28 459 


0.71 541 


9.99 209 


6 


55 


9.27 734 


9.28 527 


0.71 473 


9.99 207 


5 


56 


9.27 799 


9.28 595 


0.71 405 


9.99 204 


4 


57 


9.27 864 


9.28 662 


0.71 338 


9.99 202 


3 


58 


9.27 930 


9.28 730 


0.71 270 


9.99 200 


2 


59 
60 


9.27 995 


9.28 798 


0.71 202 


9.99 197 


1 



9.28 060 


9.28 865 


0.71 135 


9.99 195 




L. Cos. 


L. Cot. 


L. Tan. 


L. Sin. 


i 



79 



3» 



11' 



/ 


L. Sin. 


L. Tan. 


L. Cot. 


L. Cos. 






1 


9.28 060 


9.28 865 


0.71 135 


9.99 195 


60 

59 


9.28 125 


9.28 933 


0.71 067 


9.99 192 


2 


9.28 190 


9.29 000 


0.71 000 


9.99 190 


58 


3 


9.28 254 


9.29 067 


0.70 933 


9.99 187 


57 


4 


9.28 319 


9.29 134 


0.70 866 


9.99 185 


56 


5 


9.28 384 


9.29 201 


0.70 799 


9.99 182 


55 


6 


9.28 448 


9.29 268 


0.70 732 


9.99 180 


54 


7 


9.28 512 


9.29 335 


0.70 665 


9.99 177 


53 


8 


9.28 577 


9.29 402 


0.70 598 


9.99 175 


52 


9 
10 
11 


9.28 641 


9.29 468 


0.70 532 


9.99 172 


51 
50 

49 


9.28 70S 


9.29 535 


0.70 465 


9.99 170 


9.28 769 


9.29 601 


0.70 399 


9.99 167 


12 


9.28 833 


9.29 668 


0.70 332 


9.99 165 


48 


13 


9.28 896 


9.29 734 


0.70 266 


9.99 162 


47 


14 


9.28 960 


9.29 800 


0.70 200 


9.99 160 


46 


15 


9.29 024 


9.29 866 


0.70 134 


9.99 157 


45 


16 


9.29 087 


9.29 932 


0.70 068 


9.99 155 


44 


17 


9.29 150 


9.29 998 


0.70 002 


9.99 152 


43 


18 


9.29 214 


9.30 064 


0.69 936 


9.99 150 


42 


19 
20 

21 


9.29 277 


9.30 130 


0.69 870 


9.99 147 


41 
40 

39 


9.29 340 


9.30 195 


0.69 805 


9.99 145 


9.29 403 


9.30 261 


0.69 739 


9.99 142 


22 


9.29 466 


9.30 326 


0.69 674 


9.99 140 


38 


23 


9.29 529 


9.30 391 


0.69 609 


9.99 137 


37 


24 


9.29 591 


9.30 457 


0.69 543 


9.99 135 


36 


25 


9.29 654 


9.30 522 


0.69 478 


9.99 132 


35 


26 


9.29 716 


9.30 587 


0.69 413 


9.99 130 


34 


27 


9.29 779 


9.30 652 


0.69 348 


9.99 127 


33 


28 


9.29 841 


9.30 717 


0.69 283 


9.99 124 


32 


29 
30 

31 


9.29 903 


9.30 782 


0.69 218 


9.99 122 


31 
30 
29 


9.29 966 


9.30 846 


0.69 154 


9.99 119 


9.30 028 


9.30 911 


0.69 089 


9.99 117 


32 


9.30 090 


9.30 975 


0.69 025 


9.99 114 


28 


33 


9.30 151 


9.31 040 


0.68 960 


9.99 112 


27 


34 


9.30 213 


9.31 104 


0.68 896 


9.99 109 


26 


35 


9.30 275 


9.31 168 


0.68 832 


9.99 106 


25 


36 


9.30 336 


9.31 233 


0.68 767 


9.99 104 


24 


37 


9.30 398 


9.31 297 


0.68 703 


9.99 101 


23 


38 


9.30 459 


9.31 361 


0.68 639 


9.99 099 


22 


39 
40 
41 


9.30 521 


9.31 425 


0.68 575 


9.99 096 


21 
20 

19 


9.30 582 


9.31 489 


0.68 511 


9.99 093 


9.30 643 


9.31 552 


0.68 448 


9.99 091 


42 


9.30 704 


9.31 616 


0.68 384 


9.99 088 


18 


43 


9.30 765 


9.31 679 


0.68 321 


9.99 086 


17 


44 


9.30 826 


9.31 743 


0.68 257 


9.99 083 


16 


45 


9.30 887 


9.31 806 


0.68 194 


9.99 080 


15 


46 


9.30 947 


9.31 870 


0.68 130 


9.99 078 


14 


47 


9.31 008 


9.31 933 


0.68 067 


9.99 075 


13 


48 


9.31 068 


9.31 996 


0.68 004 


9.99 072 


12 


49 
50 
51 


9.31 129 


9.32 059 


0.67 941 


9.99 070 


11 
10 

9 


9.31 189 


9.32 122 


0.67 878 


9.99 067 


9.31 250 


9.32 185 


0.67 815 


9.99 064 


52 


9.31 310 


9.32 248 


0.67 752 


9.99 062 


8 


53 


9.31 370 


9.32 311 


0.67 689 


9.99 059 


7 


54 


9.31 430 


9.32 373 


0.67 627 


9.99 056 


6 


55 


9.31 490 


9.32 436 


0.67 564 


9.99 054 


5 


56 


9.31 549 


9.32 498 


0.67 502 


9.99 051 


4 


57 


9.31 609 


9.32 561 


0.67 439 


9.99 048 


3 


58 


9.31 669 


9.32 623 


0.67 377 


9.99 046 


2 


59 
60 


9.31 728 


9.32 685 


0.67 315 


9.99 043 


1 



9.31 788 


9.32 747 


0.67 253 


9.99 040 




L. Cos. 


L. Cot. 


L. Tan. 


L. Sin. 


i 



78' 



12 



39 



f 


L. Sin. 


L. Tan. 


L. Cot. 


L. Cos. 






1 


9.31 788 


9.32 747 


0.67 253 


9.99 040 


60 

59 


9.31 847 


9.32 810 


0.67 190 


9.99 038 


2 


9.31 907 


9.32 872 


0.67 128 


9.99 035 


58 


3 


9.31 966 


9.32 933 


0.67 067 


9.99 032 


57 


4 


9.32 025 


9.32 995 


0.67 005 


9.99 030 


56 


5 


9.32 084 


9.33 057 


0.66 943 


9.99 027 


55 


6 


9.32 143 


9.33 119 


0.66 881 


9.99 024 


54 


7 


9.32 202 


9.33 180 


0.66 820 


9.99 022 


53 


8 


9.32 261 


9.33 242 


0.66 758 


9.99 019 


52 


9 
10 

11 


9.32 319 


9.33 303 


0.66 697 


9.99 016 


51 
50 

49 


9.32 378 


9.33 365 


0.66 635 


9.99 013 


9.32 437 


9.33 426 


0.66 574 


9.99 011 


12 


9.32 495 


9.33 487 


0.66 513 


9.99 008 


48 


13 


9.32 553 


9.33 548 


0.66 452 


9.99 005 


47 


14 


9.32 612 


9.33 609 


0.66 391 


9.99 002 


46 


15 


9.32 670 


9.33 670 


0.66 330 


9.99 000 


45 


16 


9.32 728 


9.33 731 


0.66 269 


9.98 997 


44 


17 


9.32 786 


9.33 792 


0.66 208 


9.98 994 


43 


18 


9.32 844 


9.33 853 


0.66 147 


9.98 991 


42 


19 
20 

21 


9.32 902 


9.33 913 


0.66 087 


9.98 989 


41 
40 

39 


9.32 960 


9.33 974 


0.66 026 


9.98 986 


9.33 018 


9.34 034 


0.65 966 


9.98 983 


22 


9.33 075 


9.34 095 


0.65 905 


9.98 980 


38 


23 


9.33 133 


9.34 155 


0.65 845 


9.98 978 


37 


24 


9.33 190 


9.34 215 


0.65 785 


9.98 975 


36 


25 


9.33 248 


9.34 276 


0.65 724 


9.98 972 


35 


26 


9.33 305 


9.34 336 


0.65 664 


9.98 969 


34 


27 


9.33 362 


9.34 396 


0.65 604 


9.98 967 


33 


28 


9.33 420 


9.34 456 


0.65 544 


9.98 964 


32 


29 
30 

31 


9.33 477 


9.34 516 


0.65 484 


9.98 961 


31 
30 

29 


9.33 534 


9.34 576 


0.65 424 


9.98 958 


9.33 591 


9.34 635 


0.65 365 


9.98 955 


32 


9.33 647 


9.34 695 


0.65 305 


9.98 953 


28 


33 


9.33 704 


9.34 755 


0.65 245 


9.98 950 


27 


34 


9.33 761 


9.34 814 


0.65 186 


9.98 947 


26 


35 


9.33 818 


9.34 874 


0.65 126 


9.98 944 


25 


36 


9.33 874 


9.34 933 


0.65 067 


9.98 941 


24 


37 


9.33 931 


9.34 992 


0.65 008 


9.98 938 


23 


38 


9.33 987 


9.35 051 


0.64 949 


9.98 936 


22 


39 
40 

41 


9.34 043 


9.35 111 


0.64 889 


9.98 933 


21 
20 

19 


9.34 100 


9.35 170 


0.64 830 


9.98 930 


9.34 156 


9.35 229 


0.64 771 


9.98 927 


42 


9.34 212 


9.35 288 


0.64 712 


9.98 924 


18 


43 


9.34 268 


9.35 347 


0.64 653 


9.98 921 


17 


44 


9.34 324 


9.35 405 


0.64 595 


9.98 919 


16 


45 


9.34 380 


9.35 464 


0.64 536 


9.98 916 


15 


46 


9.34 436 


9.35 523 


0.64 477 


9.98 913 


14 


47 


9.34 491 


9.35 581 


0.64 419 


9.98 910 


13 


48 


9.34 547 


9.35 640 


0.64 360 


9.98 907 


12 


49 
50 

51 


9.34 602 


9.35 698 


0.64 302 


9.98 904 


11 
10 

9 


9.34 658 


9.35 757 


0.64 243 


9.98 901 


9.34 713 


9.35 815 


0.64 185 


9.98 898 


52 


9.34 769 


9.35 873 


0.64 127 


9.98 896 


8 


53 


9.34 824 


9.35 931 


0.64 069 


9.98 893 


7 


54 


9.34 879 


9.35 989 


0.64 Oil 


9.98 890 


6 


55 


9.34 934 


9.36 047 


0.63 953 


9.98 887 


5 


56 


9.34 989 


9.36 105 


0.63 895 


9.98 884 


4 


57 


9.35 044 


9.36 163 


0.63 837 


9.98 881 


3 


58 


9.35 099 


9.36 221 


0.63 779 


9.98 878 


2 


59 
60 


9.35 154 


9.36 279 


0.63 721 


9.98 875 


1 



9.35 209 


9.36 336 


0.63 664 


9.98 872 




L. Cos. 


L. Cot. 


L. Tan. 


L. Sin. 


i 



77' 



4o 



13 



> 


L. Sin. 


L. Tan. 


L. Cot. 


L. Cos. 






1 


9.35 209 


9.36 336 


0.63 664 


9.98 872 


60 

59 


9.35 263 


9.36 394 


0.63 606 


9.98 869 


2 


9.35 318 


9.36 452 


0.63 548 


9.98 867 


58 


3 


9.35 373 


9.36 509 


0.63 491 


9.98 864 


57 


4 


9.35 427 


9.36 566 


0.63 434 


9.98 861 


56 


5 


9.35 481 


9.36 624 


0.63 376 


9.98 858 


55 


6 


9.35 536 


9.36 681 


0.63 319 


9.98 855 


54 


7 


9.35 590 


9.36 738 


0.63 262 


9.98 852 


53 


8 


9.35 644 


9.36 795 


0.63 205 


9.98 849 


52 


9 
10 

11 


9.35 698 


9.36 852 


0.63 148 


9.98 846 


51 
50 

49 


9.35 752 


9.36 909 


0.63 091 


9.98 843 


9.35 806 


9.36 966 


0.63 034 


9.98 840 


12 


9.35 860 


9.37 023 


0.62 977 


9.98 837 


48 


13 


9.35 914 


9.37 080 


0.62 920 


9.98 834 


47 


14 


9.35 968 


9.37 137 


0.62 863 


9.98 831 


46 


15 


9.36 022 


9.37 193 


0,62 807 


9.98 828 


45 


16 


9.36 075 


9.37 250 


0. 2 750 


9.98 825 


44 


17 


9.36 129 


9.37 306 


0.62 694 


9.98 822 


43 


18 


9.36 182 


9.37 363 


0.62 637 


9.98 819 


42 


19 
20 
21 


9.36 236 


9.37 419 


0.62 581 


9.98 816 


41 
40 

39 


9.36 289 


9.37 476 


0.62 524 


9.98 813 


9.36 342 


9.37 532 


0.62 468 


9.98 810 


22 


9.36 395 


9.37 588 


0.62 412 


9.98 807 


38 


23 


9.36 449 


9.37 644 


0.62 356 


9.98 804 


37 


24 


9.36 502 


9.37 700 


0.62 300 


9.98 801 


36 


25 


9.36 555 


9.37 756 


0.62 244 


9.98 798 


35 


26 


9.36 608 


9.37 812 


0.62 188 


9.98 795 


34 


27 


9.36 660 


9.37 868 


0.62 132 


9.98 792 


33 


28 


9.36 713 


9.37 924 


0.62 076 


9.98 789 


32 


29 
30 
31 


9.36 766 


9.37 980 


0.62 020 


9.98 786 


31 
30 

29 


9.36 819 


9.38 035 


0.61 965 


9.98 783 


9.36 871 


9.38 091 


0.61 909 


9.98 780 


32 


9.36 924 


9.38 147 


0.61 853 


9.98 777 


28 


33 


9.36 976 


9.38 202 


0.61 798 


9.98 774 


27 


34 


9.37 028 


9.38 257 


0.61 743 


9.98 771 


26 


35 


9.37 081 


9.38 313 


0.61 687 


9.98 768 


25 


36 


9.37 133 


9.38 368 


0.61 632 


9.98 765 


24 


37 


9.37 185 


9.38 423 


0.61 577 


9.98 762 


23 


38 


9.37 237 


9.38 479 


0.61 521 


9.98 759 


22 


39 
40 

41 


9.37 289 


9.38 534 


0.61 466 


9.98 756 


21 
20 

19 


9.37 341 


9.38 589 


0.61 411 


9.98 753 


9.37 393 


9.38 644 


0.61 356 


9.98 750 


42 


9.37 445 


9.38 699 


0.61 301 


9.98 746 


18 


43 


9.37 497 


9.38 754 


0.61 246 


9.98 743 


17 


44 


9.37 549 


9.38 808 


0.61 192 


9.98 740 


16 


45 


9.37 600 


9.38 863 


0.61 137 


9.98 737 


15 


46 


9.37 652 


9.38 918 


0.61 082 


9.98 734 


14 


47 


9.37 703 


9.38 972 


0.61 028 


9.98 731 


13 


48 


9.37 755 


9.39 027 


0.60 973 


9.98 728 


12 


49 
50 
51 


9.37 806 


9.39 082 


0.60 918 


9.98 725 


11 
10 

9 


9.37 858 


9.39 136 


0.60 864 


9.98 722 


9.37 909 


9.39 190 


0.60 810 


9.98 719 


52 


9.37 960 


9.39 245 


0.60 755 


9.98 715 


8 


53 


9.38 Oil 


9.39 299 


0.60 701 


9.98 712 


7 


54 


9.38 062 


9.39 353 


0.60 647 


9.98 709 


6 


55 


9.38 113 


9.39 407 


0.60 593 


9.98 706 


5 


56 


9.38 164 


9.39 461 


0.60 539 


9.98 703 


4 


57 


9.38 215 


9.39 515 


0.60 485 


9.98 700 


3 


58 


9.38 266 


9.39 569 


0.60 431 


9.98 697 


2 


59 
60 


9.38 317 


9.39 623 


0.60 377 


9.98 694 


1 



9.38 368 


9.39 677 


0.60 323 


9.98 690 




L. Cos. 


L. Cot. 


L. Tan. 


L. Sin. 


/ 



76' 



14 c 



41 



1 


L. Sin. 


L. Tan. 


L. Cot. 


L. Cos. 






1 


9.38 368 


9.39 677 


0.60 323 


9.98 690 


60 

59 


9.38 418 


9.39 731 


0.60 269 


9.98 687 


2 


9.38 469 


9.39 785 


0.60 215 


9.98 684 


58 


3 


9.38 519 


9.39 838 


0.60 162 


9.98 681 


57 


4 


9.38 570 


9.39 892 


0.60 108 


9.98 678 


56 


5 


9.38 620 


9.39 945 


0.60 055 


9.98 675 


55 


6 


9.38 670 


9.39 999 


0.60 001 


9.98 671 


54 


7 


9.38 721 


9.40 052 


0.59 948 


9.98 668 


53 


8 


9.38 771 


9.40 106 


0.59 894 


9.98 665 


52 


9 
10 

11 


9.38 821 


9.40 159 


0.59 841 


9.98 662 


51 
50 

49 


9.38 871 


9.40 212 


0.59 788 


9.98 659 


9.38 921 


9.40 266 


0.59 734 


9.98 656 


12 


9.38 971 


9.40 319 


0.59 681 


9.98 652 


48 ... 


13 


9.39 021 


9.40 372 


0.59 628 


9.98 649 


47 


14 


9.39 071 


9.40 425 


0.59 575 


9.98 646 


46 


15 


9.39 121 


9.40 478 


0.59 522 


9.98 643 


45 


16 


9.39 170 


9.40 531 


0.59 469 


9.98 640 


44 


17 


9.39 220 


9.40 584 


0.59 416 


9.98 636 


43 


18 


9.39 270 


9.40 636 


0.59 364 


9.98 633 


42 


19 
20 
21 


9.39 319 


9.40 689 


0.59 311 


9.98 630 


41 
40 

39 


9.39 369 


9.40 742 


0.59 258 


9.98 627 


9.39 418 


9.40 795 


0.59 205 


9.98 623 


22 


9.39 467 


9.40 847 


0.59 153 


9.98 620 


38 


23 


9.39 517 


9.40 900 


0.59 100 


9.98 617 


37 


24 


9.39 566 


9.40 952 


0.59 048 


9.98 614 


36 


25 


9.39 615 


9.41 005 


0.58 995 


9.98 610 


35 


26 


9.39 664 


9.41 057 


0.58 943 


9.98 607 


34 


27 


9.39 713 


9.41 109 


0.58 891 


9.98 604 


33 


28 


9.39 762 


9.41 161 


0.58 839 


9.98 601 


32 


29 
30 

31 


9.39 811 


9.41 214 


0.58 786 


9.98 597 


31 
30 
29 


9.39 860 


9.41 266 


0.58 734 


9.98 594 


9.39 909 


9.41 318 


0.58 682 


9.98 591 


32 


9.39 958 


9.41 370 


0.58 630 


9.98 588 


28 


33 


9.40 006 


9.41 422 


0.58 578 


9.98 584 


27 


34 


9.40 055 


9.41 474 


0.58 526 


9.98 581 


26 


35 


9.40 103 


9.41 526 


0.58 474 


9.98 578 


25 


36 


9.40 152 


9.41 578 


0.58 422 


9.98 574 


24 


37 


9.40 200 


9.41 629 


0.58 371 


9.98 571 


23 


38 


9.40 249 


9.41 681 


0.58 319 


9.98 568 


22 


39 
40 

41 


9.40 297 


9.41 733 


0.58 267 


9.98 565 


21 
20 

19 


9.40 346 


9.41 784 


0.58 216 


9.98 561 


9.40 394 


9.41 836 


0.58 164 


9.98 558 


42 


9.40 442 


9.41 887 


0.58 113 


9.98 555 


18 


43 


9.40 490 


9.41 939 


0.58 061 


9.98 551 


17 


44 


9.40 538 


9.41 990 


0.58 010 


9.98 548 


16 


45 


9.40 586 


9.42 041 


0.57 959 


9.98 545 


15 


46 


9.40 634 


9.42 093 


0.57 907 


9.98 541 


14 


47 


9.40 682 


9.42 144 


0.57 856 


9.98 538 


13 


48 


9.40 730 


9.42 195 


0.57 805 


9.98 535 


12 


49 
50 

51 


9.40 778 


9.42 246 


0.57 754 


9.98 531 


11 
10 

9 


9.40 825 


9.42 297 


0.57 703 


9.98 528 


9.40 873 


9.42 348 


0.57 652 


9.98 525 


52 


9.40 921 


9.42 399 


0.57 601 


9.98 521 


8 


53 


9.40 968 


9.42 450 


0.57 550 


9.98 518 


7 


54 


9.41 016 


9.42 501 


0.57 499 


9.98 515 


6 


55 


9.41 063 


9.42 552 


0.57 448 


9.98 511 


5 


m 


9.41 111 


9.42 603 


0.57 397 


9.98 508 


4 


57 


9.41 158 


9.42 653 


0.57 347 


9.98 505 


3 


58 


9.41 205 


9.42 704 


0.57 296 


9.98 501 


2 


59 
60 


9.41 252 


9.42 755 


0.57 245 


9.98 498 


1 



9.41 300 


9.42 805 


0.57 195 


9.98 494 




L. Cos. 


L. Cot. 


L. Tan. 


L. Sin. 


/ 



75' 



42 





1 


L. Sin. 


L. Tan. 


L. Cot. 


L. Cos. 




9.41 300 


9.42 805 


0.57 195 


9.98 494 


60 

59 


9.41 347 


9.42 856 


0.57 144 


9.98 491 


2 


9.41 394 


9.42 906 


0.57 094 


9.98 488 


58 


3 


9.41 441 


9.42 957 


0.57 043 


9.98 484 


57 


4 


9.41 488 


9.43 007 


0.56 993 


9.98 481 


56 


5 


9.41 535 


9.43 057 


0.56 943 


9.98 477 


55 


6 


9.41 582 


9.43 108 


0.56 892 


9.98 474 


54 


7 


9.41 628 


9.43 158 


0.56 842 


9.98 471 


53 


8 


9.41 675 


9.43 208 


0.56 792 


9.98 467 


52 


9 
10 

11 


9.41 722 


9.43 258 


0.56 742 


9.98 464 


51 
50 

49 


9.41 768 


9.43 308 


0.56 692 


9.98 460 


9.41 815 


9.43 358 


0.56 642 


9.98 457 


12 


9.41 861 


9.43 408 


0.56 592 


9.98 453 


48 


13 


9.41 908 


9.43 458 


0.56 542 


9.98 450 


47 


14 


9.41 954 


9.43 508 


0.56 492 


9.98 447 


46 


15 


9.42 001 


9.43 558 


0.56 442 


9.98 443 


45 


16 


9.42 047 


9.43 607 


0.56 393 


9.98 440 


44 


17 


9.42 093 


9.43 657 


0.56 343 


9.98 436 


43 


18 


9.42 140 


9.43 707 


0.56 293 


9.98 433 


42 


19 
20 

21 


9.42 186 


9.43 756 


0.56 244 


9.98 429 


41 
40 

39 


9.42 232 


9.43 806 


0.56 194 


9.98 426 


9.42 278 


9.43 855 


0.56 145" 


9.98 422 


22 


9.42 324 


9.43 905 


0.56 095 


9.98 419 


38 


23 


9.42 370 


9.43 954 


0.56 046 


9.98 415 


37 


24 


9.42 416 


9.44 004 


0.55 996 


9.98 412 


36 


25 


9.42 461 


9.44 053 


0.55 947 


9.98 409 


35 


26 


9.42 507 


9.44 102 


0.55 898 


9.98 405 


34 


27 


9.42 553 


9.44 151 


0.55 849 


9.98 402 


33 


28 


9.42 599 


9.44 201 


0.55 799 


9.98 398 


32 


29 
30 

31 


9.42 644 


9.44 250 


0.55 750 


9.98 395 


31 
30 

29 


9.42 690 


9.44 299 


0.55 701 


9.98 391 


9.42 735 


9.44 348 


0.55 652 


9.98 388 


32 


9.42 781 


9.44 397 


0.55 603 


9.98 384 


28 


33 


9.42 826 


9.44 446 


0.55 554 


9.98 381 


27 


34 


9.42 872 


9.44 495 


0.55 505 


9.98 377 


26 


35 


9.42 917 


9.44 544 


0.55 456 


9.98 373 


25 


36 


9.42 962 


9.44 592 


0.55 408 


9.98 370 


24 


37 


9.43 008 


9.44 641 


0.55 359 


9.98 366 


23 


38 


9.43 053 


9.44 690 


0.55 310 


9.98 363 


22 


39 
40 

41 


9.43 098 


9.44 738 


0.55 262 


9.98 359 


21 
20 

19 


9.43 143 


9.44 787 


0.55 213 


9.98 356 


9.43 188 


9.44 836 


0.55 164 


9.98 352 


42 


9.43 233 


9.44 884 


0.55 116 


9.98 349 


18 


43 


9.43 278 


9.44 933 


0.55 067 


9.98 345 


17 


44 


9.43 323 


9.44 981 


0.55 019 


9.98 342 


16 


45 


9.43 367 


9.45 029 


0.54 971 


9.98 338 


15 


46 


9.43 412 


9.45 078 


0.54 922 


9.98 334 


14 


47 


9.43 457 


9.45 126 


0.54 874 


9.98 331 


13 


48 


9.43 502 


9.45 174 


0.54 826 


9.98 327 


12 


49 
50 

51 


9.43 546 


9.45 222 


0.54 778 


9.98 324 


11 
10 

9 


9.43 591 


9.45 271 


0.54 729 


9.98 320 


9.43 635 


9.45 319 


0.54 681 


9.98 317 


52 


9.43 680 


9.45 367 


0.54 633 


9.98 313 


8 


53 


9.43 724 


9.45 415 


0.54 585 


9.98 309 


7 


54 


9.43 769 


9.45 463 


0.54 537 


9.98 306 


6 


55 


9.43 813 


9.45 511 


0.54 489 


9.98 302 


5 


56 


9.43 857 


9.45 559 


0.54 441 


9.98 299 


4 


57 


9.43 901 


9.45 606 


0.54 394 


9.98 295 


3 


58 


9.43 946 


9.45 654 


0.54 346 


9.98 291 


2 


59 
60 


9.43 990 


9.45 702 


0.54 298 


9.98 288 


1 



9.44 034 


9.45 750 


0.54 250 


9.98 284 




L. Cos. 


L. Cot. 


L. Tan. 


L. Sin. 


/ 



74 c 



16 



43 



f 


L. Sin. 


L. Tan. 


L. Cot. 


L. Cos. 






1 


9.44 034 


9.45 750 


0.54 250 


9.98 284 


60 

59 


9.44 078 


9.45 797 


0.54 203 


9.98 281 


2 


9.44 122 


9.45 845 


0.54 155 


9.98 277 


58 


3 


9.44 166 


9.45 892 


0.54 108 


9.98 273 


57 


4 


9.44 210 


9.45 940 


0.54 060 


9.98 270 


56 


5 


9.44 253 


9.45 987 


0.54 013 


9.98 266 


55 


6 


9.44 297 


9.46 035 


0.53 965 


9.98 262 


54 


7 


9.44 341 


9.46 082 


0.53 918 


9.98 259 


53 


8 


9.44 385 


9.46 130 


0.53 870 


9.98 255 


52 


9 
10 

11 


9.44 428 


9.46 177 


0.53 823 


9.98 251 


51 
50 

49 


9.44 472 


9.46 224 


0.53 776 


9.98 248 


9.44 516 


9.46 271 


0.53 729 


9.98 244 


12 


9.44 559 


9.46 319 


0.53 681 


9.98 240 


48 


13 


9.44 602 


9.46 366 


0.53 6:34 


9.98 237 


47 


14 


9.44 646 


9.46 413 


0.53 587 


9.98 233 


46 


15 


9.44 689 


9.46 460 


0.53 540 


9.98 229 


45 


16 


9.44 733 


9.46 507 


0.53 493 


9.98 226 


44 


17 


9.44 776 


9.46 554 


0.53 446 


9.98 222 


43 


18 


9.44 819 


9.46 601 


0.53 399 


9.98 218 


42 


19 
20 

21 


9.44 862 


9.46 648 


0.53 352 


9.98 215 


41 
40 

39 


9.44 905 


9.46 694 


0.53 306 


9.98 211 


9.44 948 


9.46 741 


0.53 259 


9.98 207 


22 


9.44 992 


9.46 788 


0.53 212 


9.98 204 


38 


23 


9.45 035 


9.46 835 


0.53 165 


9.98 200 


37 


24 


9.45 077 


9.46 881 


0.53 119 


9.98 196 


36 


25 


9.45 120 


9.46 928 


0.53 072 


9.98 192 


35 


26 


9.45 163 


9.46 975 


0.53 025 


9.98 189 


34 


27 


9.45 206 


9.47 021 


0.52 979 


9.98 185 


33 


28 


9.45 249 


9.47 068 


0.52 932 


9.98 181 


32 


29 
30 

31 


9.45 292 


9.47 114 


0.52 886 


9.98 177 


31 
30 

29 


9.45 3:34 


9.47 160 


0.52 840 


9.98 174 


9.45 377 


9.47 207 


0.52 793 


9.98 170 


32 


9.45 419 


9.47 253 


0.52 747 


9.98 166 


28 


33 


9.45 462 


9.47 299 


0.52 701 


9.98 162 


27 


34 


9.45 504 


9.47 346 


0.52 654 


9.98 159 


26 


35 


9.45 547 


9.47 392 


0.52 608 


9.98 155 


25 


36 


9.45 589 


9.47 438 


0.52 562 


9.98 151 


24 


37 


9.45 632 


9.47 484 


0.52 516 


9.98 147 


23 


38 


9.45 674 


9.47 530 


0.52 470 


9.98 144 


22 


39 
40 

41 


9.45 716 


9.47 576 


0.52 424 


9.98 140 


21 
20 

19 


9.45 758 


9.47 622 


0.52 378 


9.98 136 


9.45 801 


9.47 668 


0.52 332 


9.98 132 


42 


9.45 843 


9.47 714 


0.52 286 


9.98 129 


18 


43 


9.45 885 


9.47 760 


0.52 240 


9.98 125 


17 


44 


9.45 927 


9.47 806 


0.52 194 


9.98 121 


16 


45 


9.45 969 


9.47 852 


0.52 148 


9.98 117 


15 


46 


9.46 Oil 


9.47 897 


0.52 103 


9.98 113 


14 


47 


9.46 053 


9.47 943 


0.52 057 


9.98 110 


13 


48 


9.46 095 


9.47 989 


0.52 Oil 


9.98 106 


12 


49 
50 

51 


9.46 136 


9.48 035 


0.51 965 


9.98 102 


11 

10 

9 


9.46 178 


9.48 080 


0.51 920 


9.98 098 


9.46 220 


9.48 126 


0.51 374 


9.98 094 


52 


9.46 262 


9.48 171 


0.51 829 


9.98 090 


8 


53 


9.46 303 


9.48 217 


0.51 783 


9.98 087 


7 


54 


9.46 345 


9.48 262 


0.51 738 


9.98 083 


6 


55 


9.46 386 


9.48 307 


0.51 693 


9.98 079 


5 


56 


9.46 428 


9.48 353 


0.51 647 


9.98 075 


4 


57 


9.46 469 


9.48 398 


0.51 602 


9.98 071 


3 


•58 


9.46 511 


9.48 443 


0.51 557 


9.98 067 


2 


59 
60 


9.46 552 


9.48 489 


0.51 511 


9.98 063 


1 



9.46 594 


9.48 534 


0.51 466 


9.98 060 


1 


L. Cos. 


L, Cot. 


L. Tan. 


L. Sin. 


/ 



73 



44 



17 



t 


L. Sin. 


L. Tan. 


L. Cot. 


L. Cos. 






1 


9.46 594 


9.48 534 


0.51 466 


9.98 060 


60 

59 


9.46 635 


9.48 579 


0.51 421 


9.98 056 


2 


9.46 676 


9.48 624 


0.51 376 


9.98 052 


58 


3 


9.46 717 


9.48 669 


0.51 331 


9.98 048 


57 


4 


9.46 758 


9.48 714 


0.51 286 


9.98 044 


56 


5 


9.46 800 


9.48 759 


0.51 241 


9.98 040 


55 


6 


9.46 841 


9.48 804 


0.51 196 


9.98 036 


54 


7 


9.46 882 


9.48 849 


0.51 151 


9.98 032 


53 


8 


9.46 923 


9.48 894 


0.51 106 


9.98 029 


52 


9 
10 

11 


9.46 964 


9.48 939 


0.51 061 


9.98 025 


51 
50 

49 


9.47 005 


9.48 984 


0.51 016 


9.98 021 


9.47 045 


9.49 029 


0.50 971 


9.98 017 


12 


9.47 086 


9.49 073 


0.50 927 


9.98 013 


48 


13 


9.47 127 


9.49 118 


0.50 882 


9.98 009 


47 


14 


9.47 168 


9.49 163 


0.50 837 


9.98 005 


46 


15 


9.47 209 


9.49 207 


0.50 793 


9.98 001 


45 


16 


9.47 249 


9.49 252 


0.50 748 


9.97 997 


44 


17 


9.47 290 


9.49 296 


0.50 704 


9.97 993 


43 


18 


9.47 330 


9.49 341 


0.50 659 


9.97 989 


42 


19 
20 
21 


9.47 371 


9.49 385 


0.50 615 


9.97 986 


41 
40 

39 


9.47 411 


9.49 430 


0.50 570 


9.97 982 


9.47 452 


9.49 474 


0.50 526 


9.97 978 


22 


9.47 492 


9.49 519 


0.50 481 


9.97 974 


38 


23 


9.47 533 


9.49 563 


0.50 437 


9.97 970 


37 


24 


9.47 573 


9.49 607 


0.50 393 


9.97 966 


36 


25 


9.47 613 


9.49 652 


0.50 348 


9.97 962 


35 


26 


9.47 654 


9.49 696 


0.50 304 


9.97 958 


34 


27 


9.47 694 


9.49 740 


0.50 260 


9.97 954 


33 


28 


9.47 734 


9.49 784 


0.50 216 


9.97 950 


32 


29 
30 
31 


9.47 774 


9.49 828 


0.50 172 


9.97 946 


31 
30 

29 


9.47 814 


9.49 872 


0.50 128 


9.97 942 


9.47 854 


9.49 916 


0.50 084 


9.97 938 


32 


9.47 894 


9.49 960 


0.50 040 


9.97 934 


28 


33 


9.47 934 


9.50 04 


0.49 996 


9.97 930 


27 


34 


9.47 974 


9.50 048 


0.49 952 


9.97 926 


26 


35 


9.48 014 


9.50 092 


0.49 908 


9.97 922 


25 


36 


9.48 054 


9.50 136 


0.49 864 


9.97 918 


24 


37 


9.48 094 


9.50 180 


0.49 820 


9.97 914 


23 


38 


9.48 133 


9.50 223 


0.49 777 


9.97 910 


22 


39 
40 
41 


9.48 173 


9.50 267 


0.49 733 


9.97 906 


21 
20 

19 


9.48 213 


9.50 311 


0.49 689 


9.97 902 


9.48 252 


9.50 355 


0.49 645 


9.97 898 


42 


9.48 292 


9.50 398 


0.49 602 


9.97 894 


18 


43 


9.48 332 


9.50 442 


0.49 558 


9.97 890 


17 


44 


9.48 371 


9.50 485 


0.49 515 


9.97 886 


16 


45 


9.48 411 


9.50 529 


0.49 471 


9.97 882 


15 


46 


9.48 450 


9.50 572 


0.49 428 


9.97 878 


14 


47 


9.48 490 


9.50 616 


0.49 384 


9.97 874 


13 


48 


9.48 529 


9.50 659 


0.49 341 


9.97 870 


12 


49 
50 
51 


9.48 568 


9.50 703 


0.49 297 


9.97 866 


11 
10 

9 


9.48 607 


9.50 746 


0.49 254 


9.97 861 


9.48 647 


9.50 789 


0.49 211 


9.97 857 


52 


9.48 686 


9.50 833 


0.49 167 


9.97 853 


8 


53 


9.48 725 


9.50 876 


0.49 124 


9.97 849 


7 


54 


9.48 764 


9.50 919 


0.49 081 


9.97 845 


6 


55 


9.48 803 


9.50 962 


0.49 038 


9.97 841 


5 


56 


9.48 842 


9.51 005 


0.48 995 


9.97 837 


4 


57 


9.48 881 


9.51 048 


0.48 952 


9.97 833 


3 


58 


9.48 920 


9.51 092 


0.48 908 


9.97 829 


2 


59 
60 


9.48 959 


9.51 135 


0.48 865 


9.97 825 


1 



9.48 998 


9.51 178 


0.48 822 


9.97 821 




L. Cos. 


L. Cot. 


L. Tan. 


L. Sin. 


t 



72 c 



18 c 



45 



t 


L. Sin. 


L. Tan. 


L. Cot. 


L. Cos. 






1 


9.48 998 


9.51 178 


0.48 822 


9.97 821 


60 

59 


9.49 037 


9.51 221 


0.48 779 


9.97 817 


2 


9.49 076 


9.51 264 


0.48 736 


9.97 812 


58 


3 


9.49 115 


9.51 306 


0.48 694 


9.97 808 


57 


4 


9.49 153 


9.51 349 


0.48 651 


9.97 804 


56 


5 


9.49 192 


9.51 392 


0.48 608 


9.97 800 


55 


6 


9.49 231 


9.51 435 


0.48 565 


9.97 796 


54 


7 


9.49 269 


9.51 478 


0.48 522 


9.97 792 


53 


8 


9.49 308 


9.51 520 


0.48 480 


9.97 788 


52 


9 
10 

11 


9.49 347 


9.51 563 


0.48 437 


9.97 784 


51 
50 

49 


9.49 385 


9.51 606 


0.48 394 


9.97 779 


9.49 424 


9.51 648 


0.48 352 


9.97 775 


12 


9.49 462 


9.51 691 


0.48 309 


9.97 771 


48 


13 


9.49 500 


9.51 734 


0.48 266 


9.97 767 


47 


14 


9.49 539 


9.51 776 


0.48 224 


9.97 763 


46 


15 


9.49 577 


9.51 819 


0.48 181 


9.97 759 


45 


16 


9.49 615 


9.51 861 


0.48 139 


9.97 754 


44 


17 


9.49 654 


9.51 903 


0.48 097 


9.97 750 


43 


18 


9.49 692 


9.51 946 


0.48 054 


9.97 746 


42 


19 
20 

21 


9.49 730 


9.51 988 


0.48 012 


9.97 742 


41 
40 

39 


9.49 768 


9.52 031 


0.47 969 


9.97 738 


9.49 806 


9.52 073 


0.47 927 


9.97 734 


22 


9.49 844 


9.52 115 


0.47 885 


9.97 729 


38 


23 


9.49 882 


9.52 157 


0.47 843 


9.97 725 


37 


21 


9.49 920 


9.52 200 


0.47 800 


9.97 721 


36 


25 


9.49 958 


9.52 242 


0.47 758 


9.97 717 


35 


26 


9.49 996 


9.52 284 


0.47 716 


9.97 713 


34 


27 


9.50 034 


9.52 326 


0.47 674 


9.97 708 


33 


28 


9.50 072 


9.52 368 


0.47 632 


9.97 704 


32 


29 
30 

31 


9.50 110 


9.52 410 


0.47 590 


9.97 700 


31 
30 

29 


9.50 148 


9.52 452 


0.47 548 


9.97 696 


9.50 185 


9.52 494 


0.47 506 


9.97 691 


32 


9.50 223 


9.52 536 


0.47 464 


9.97 687 


28 


33 


9.50 261 


9.52 578 


0.47 422 


9.97 683 


27 


34 


9.50 298 


9.52 620 


0.47 380 


9.97 679 


26 


35 


9.50 336 


9.52 661 


0.47 339 


9.97 674 


25 


36 


9.50 374 


9.52 703 


0.47 297 


9.97 670 


24 


37 


9.50 411 


9.52 745 


0.47 255 


9.97 666 


23 


38 


9.50 449 


9.52 787 


0.47 213 


9.97 662 


22 


39 
40 

41 


9.50 486 


9.52 829 


0.47 171 


9.97 657 


21 
20 

19 


9.50 523 


9.52 870 


0.47 130 


9.97 653 


9.50 561 


9.52 912 


0.47 088 


9.97 649 


42 


9.50 598 


9.52 953 


0.47 047 


9.97 645 


18 


43 


9.50 635 


9.52 995 


0.47 005 


9.97 640 


17 


44 


9.50 673 


9.53 037 


0.46 963 


9.97 636 


16 


45 


9.50 710 


9.53 078 


0.46 922 


9.97 632 


15 


46 


9.50 747 


9.53 120 


0.46 880 


9.97 628 


14 


47 


9.50 784 


9.53 161 


0.46 839 


9.97 623 


13 


48 


9.50 821 


9.53 202 


0.46 798 


9.97 619 


12 


49 
50 
51 


9.50 858 


9.53 244 


0.46 756 


9.97 615 


11 
10 

9 


9.50 896 


9.53 285 


0.46 715 


9.97 610 


9.50 933 


9.53 327 


0.46 673 


9.97 606 


52 


9.50 970 


9.53 368 


0.46 632 


9.97 602 


8 


53 


9.51 007 


9.53 409 


0.46 591 


9.97 597 


7 


54 


9.51 043 


9.53 450 


0.46 550 


9.97 593 


6 


55 


9.51 080 


9.53 492 


0.46 508 


9.97 589 


5 


56 


9.51 117 


9.53 533 


0.46 467 


9.97 584 


4 


57 


9.51 154 


9.53 574 


0.46 426 


9.97 580 


3 


58 


9.51 191 


9.53 615 


0.46 385 


9.97 576 


2 


59 
60 


9.51 227 


9.53 656 


0.46 344 


9.97 571 


1 



9.51 264 


9.53 697 


0.46 303 


9.97 567 




L. Cos. 


L. Cot. 


L. Tan. 


L. Sin. 


t 



7V 



46 



19 



/ 


L. Sin. 


L. Tan. 


L. Cot. 


L. Cos. 






1 


9.51 264 


9.53 697 


0.46 303 


9.97 567 


60 

59 


9.51 301 


9.53 738 


0.46 262 


9.97 563 


2 


9.51 338 


9.53 779 


0.46 221 


9.97 558 


58 


3 


9.51 374 


9.53 820 


0.46 180 


9.97 554 


57 


4 


9.51 411 


9.53 861 


0.46 139 


9.97 550 


56 


5 


9.51 447 


9.53 902 


0.46 098 


9.97 545 


55 


6 


9.51 484 


9.53 943 


0.46 057 


9.97 541 


54 


7 


9.51 520 


9.53 984 


0.46 016 


9.97 536 


53 


8 


9.51 557 


9.54 025 


0.45 975 


9.97 532 


52 


9 
10 

11 


9.51 593 


9.54 065 


0.45 935 


9.97 528 


51 
50 

49 


9.51 629 


9.54 106 


0.45 894 


9.97 523 


9.51 666 


9.54 147 


0.45 853 


9.97 519 


12 


9.51 702 


9.54 187 


0.45 813 


9.97 515 


48 


13 


9.51 738 


9.54 228 


0.45 772 


9.97 510 


47 


14 


9.51 774 


9.54 269 


0.45 731 


9.97 506 


46 


15 


9.51 811 


9.54 309 


0.45 691 


9.97 501 


45 


16 


9.51 847 


9.54 350 


0.45 650 


9.97 497 


44 


17 


9.51 883 


9.54 390 


0.45 610 


9.97 492 


43 


18 


9.51 919 


9.54 431 


0.45 569 


9.97 488 


42 


19 
20 
21 


9.51 955 


9.54 471 


0.45 529 


9.97 484 


41 
40 

39 


9.51 991 


9.54 512 


0.45 488 


9.97 479 


9.52 027 


9.54 552 


0.45 448 


9.97 475 


22 


9.52 063 


9.54 593 


0.45 407 


9.97 470 


38 


23 


9.52 099 


9.54 633 


0.45 367 


9.97 466 


37 


24 


9.52 135 


9.54 673 


0.45 327 


9.97 461 


36 


25 


9.52 171 


9.54 714 


0.45 286 


9.97 457 


35 


26 


9.52 207 


9.54 754 


0.45 246 


9.97 453 


34 


27 


9.52 242 


9.54 794 


0.45 206 


9.97 448 


33 


28 


9.52 278 


9.54 835 


0.45 165 


9.97 444 


32 


29 
30 

31 


9.52 314 


9.54 875 


0.45 125 


9.97 439 


31 
30 

29 


9.52 350 


9.54 915 


0.45 085 


9.97 435 


9.52 385 


9.54 955 


0.45 045 


9.97 430 


32 


9.52 421 


9.54 995 


0.45 005 


9.97 426 


28 


33 


9.52 456 


9.55 035 


0.44 965 


9.97 421 


27 


34 


9.52 492 


9.55 075 


0.44 925 


9.97 417 


26 


35 


9.52 527 


9.55 115 


0.44 885 


9.97 412 


25 


36 


9.52 563 


9.55 155 


0.44 845 


9.97 408 


24 


37 


9.52 598 


9.55 195 


0.44 805 


9.97 403 


23 


38 


9.52 634 


9.55 235 


0.44 765 


9.97 399 


22 


39 
40 

41 


9.52 669 


9.55 275 


0.44 725 


9.97 394 


21 
20 

19 


9.52 705 


9.55 315 


0.44 685 


9.97 390 


9.52 740 


9.55 355 


0.44 645 


9.97 385 


42 


9.52 775 


9.55 395 


0.44 605 


9.97 381 


18 


43 


9.52 811 


9.55 434 


0.44 566 


9.97 376 


17 


44 


9.52 846 


9.55 474 


0.44 526 


9.97 372 


16 


45 


9.52 881 


9.55 514 


0.44 486 


9.97 367 


15 


46 


9.52 916 


9.55 554 


0.44 446 


9.97 363 


14 


47 


9.52 951 


9.55 593 


0.44 407 


9.97 358 


13 


48 


9.52 986 


9.55 633 


0.44 367 


9.97 353 


12 


49 
50 
51 


9.53 021 


9.55 673 


0.44 327 


9.97 349 


11 
10 

9 


9.53 056 


9.55 712 


0.44 288 


9.97 344 


9.53 092 


9.55 752 


0.44 248 


9.97 340 


52 


9.53 126 


9.55 791 


0.44 209 


9.97 335 


8 


53 


9.53 161 


9.55 831 


0.44 169 


9.97 331 


7 


54 


9.53 196 


9.55 870 


0.44 130 


9.97 326 


6 


55 


9.53 231 


9.55 910 


0.44 090 


9.97 322 


5 


56 


9.53 266 


9.55 949 


0.44 051 


9.97 317 


4 


57 


9.53 301 


9.55 989 


0.44 Oil 


9.97 312 


3 


58 


9.53 336 


9.56 028 


0.43 972 


9.97 308 


2 


59 
60 


9.53 370 


9.56 067 


0.43 933 


9.97 303 


1 



9.53 405 


9.56 107 


0.43 893 


9.97 299 




L. Cos. 


L. Cot. 


L. Tan. 


L. Sin. 


/ 



70 



20 



47 



/ 


L. Sin. 


L. Tan. 


L. Cot. 


L. Cos. 






1 


9.53 405 


9.56 107 


0.43 893 


9.97 299 


60 

59 


9.53 440 


9.56 146 


0.43 854 


9.97 294 


2 


9.53 475 


9.56 185 


0.43 815 


9.97 289 


58 


3 


9.53 509 


9.56 224 


0.43 776 


9.97 285 


57 


4 


9.53 544 


9.56 264 


0.43 736 


9.97 280 


56 


5 


9.53 578 


9.56 303 


0.43 697 


9.97 276 


55 


6 


9.53 613 


9.56 342 


0.43 658 


9.97 271 


54 


7 


9.53 647 


9.56 381 


0.43 619 


9.97 266 


53 


8 


9.53 682 


9.56 420 


0.43 580 


9.97 262 


52 


9 
10 

11 


9.53 716 


9.56 459 


0.43 541 


9.97 257 


51 
50 

49 


9.53 751 


9.56 498 


0.43 502 


9.97 252 


9.53 785 


9.56 537 


0.43 463 


9.97 248 


12 


9.53 819 


9.56 576 


0.43 424 


9.97 243 


48 


13 


9.53 854 


9.56 615 


0.43 385 


9.97 238 


47 


14 


9.53 888 


9.56 654 


0.43 346 


9.97 234 


46 


15 


9.53 922 


9.56 693 


0.43 307 


9.97 229 


45 


16 


9.53 957 


9.56 732 


0.43 268 


9.97 224 


44 


17 


9.53 991 


9.56 771 


0.43 229 


9.97 220 


43 


18 


9.54 025 


9.56 810 


0.43 190 


9.97 215 


42 


19 
20 

21 


9.54 059 


9.56 849 


0.43 151 


9.97 210 


41 
40 

39 


9.54 093 


9.56 887 


0.43 113 


9.97 206 


9.54 127 


9.56 926 


0.43 074 


9.97 201 


22 


9.54 161 


9.56 965 


0.43 035 


9.97 196 


38 


23 


9.54 195 


9.57 004 


0.42 996 


9.97 192 


37 


24 


9.54 229 


9.57 042 


0.42 958 


9.97 187 


36 


25 


9.54 263 


9.57 081 


0.42 919 


9.97 182 


35 


26 


9.54 297 


9.57 120 


0.42 880 


9.97 178 


34 


27 


9.54 331 


9.57 158 


0.42 842 


9.97 173 


33 


28 


9.54 365 


9.57 197 


0.42 803 


9.97 168 


32 


29 
30 

31 


9.54 399 


9.57 235 


0.42 765 


9.97 163 


31 
30 

29 


9.54 433 


9.57 274 


0.42 726 


9.97 159 


9.54 466 


9.57 312 


0.42 688 


9.97 154 


32 


9.54 500 


9.57 351 


0.42 649 


9.97 149 


28 


33 


9.54 534 


9.57 389 


0.42 611 


9.97 145 


27 


34 


9.54 567 


9.57 428 


0.42 572 


9.97 140 


26 


35 


9.54 601 


9.57 466 


0.42 534 


9.97 135 


25 


36 


9.54 635 


9.57 504 


0.42 496 


9.97 130 


24 


37 


9.54 668 


9.57 543 


0.42 457 


9.97 126 


23 


38 


9.54 702 


9.57 581 


0.42 419 


9.97 121 


22 


39 
40 

41 


9.54 735 


9.57 619 


0.42 381 


9.97 116 


21 
20 

19 


9.54 769 


9.57 658 


0.42 342 


9.97 111 


9.54 802 


9.57 696 


0.42 304 


9.97 107 


42 


9.54 836 


9.57 734 


0.42 266 


9.97 102 


18 


43 


9.54 869 


9.57 772 


0.42 228 


9.97 097 


17 


44 


9.54 903 


9.57 810 


0.42 190 


9.97 092 


16 


45 


9.54 936 


9.57 849 


0.42 151 


9.97 087 


15 


46 


9.54 969 


9.57 887 


0.42 113 


9.97 083 


14 


47 


9.55 003 


9.57 925 


0.42 075 


9.97 078 


13 


48 


9.55 036 


9.57 963 


0.42 037 


9.97 073 


12 


49 
50 

51 


9.55 069 


9.58 001 


0.41 999 


9.97 068 


11 
10 

9 


9.55 102 


9.58 039 


0.41 961 


9.97 063 


9.55 136 


9.58 077 


0.41 923 


9.97 059 


52 


9.55 169 


9.58 115 


0.41 885 


9.97 054 


8 


53 


9.55 202 


9.58 153 


0.41 847 


9.97 049 


7 


54 


9.55 235 


9.58 191 


0.41 809 


9.97 044 


6 


55 


9.55 268 


9.58 229 


0.41 771 


9.97 039 


5 


56 


9.55 301 


9.58 267 


0.41 733 


9.97 035 


4 


57 


9.55 334 


9.58 304 


0.41 696 


9.97 030 


3 


58 


9.55 367 


9.58 342 


0.41 658 


9.97 025 


2 


59 
60 


9.55 400 


9.58 380 


0.41 620 


9.97 020 


1 



9.55 433 


9.58 418 


0.41 582 


9.97 015 

"^usTnT" 




L, Cos. 


L. Cot. 


L. Tan. 


i 



69' 



4 8 



2V 



t 


L. Sin. 


L. Tan. 


L. Cot. 


L. Cos. 






1 


9.55 433 


9.58 418 


0.41 582 


9.97 015 


60 

59 


9.55 4(56 


9.58 455 


0.41 545 


9.97 010 


2 


9.55 499 


9.58 493 


0.41 507 


9.97 005 


58 


3 


9.55 532 


9.58 531 


0.41 469 


9.97 001 


57 


4 


9.55 564 


9.58 569 


0.41 431 


9.96 996 


56 


5 


9.55 597 


9.58 606 


0.41 394 


9.96 991 


55 


6 


9.55 630 


9.58 644 


0.41 356 


9.96 986 


54 


7 


9.55 663 


9.58 681 


0.41 319 


9.96 981 


53 


8 


9.55 695 


9.58 719 


0.41 281 


9.96 976 


52 


9 
10 

11 


9.55 728 


9.58 757 


0.41 243 


9.96 971 


51 
50 

49 


9.55 761 


9.58 794 


0.41 206 


9.96 966 


9.55 793 


9.58 832 


0.41 168 


9.96 962 


12 


9.55 826 


9.58 869 


0.41 131 


9.96 957 


48 


13 


9.55 858 


9.58 907 


0.41 093 


9.96 952 


47 


14 


9.55 891 


9.58 944 


0.41 056 


9.96 947 


46 


15 


9.55 923 , 


9.58 981 


0.41 019 


9.96 942 


45 


16 


9.55 956 


9.59 019 


0.40 981 


9.96 937 


44 


17 


9.55 988 


9.59 056 


0.40 944 


9.96 932 


43 


18 


9.56 021 


9.59 094 


0.40 906 


9.96 927 


42 


19 
20 
21 


9.56 053 


9.59 131 


0.40 869 


9.96 922 


41 
40 

39 


9.56 085 


9.59 168 


0.40 832 


9.96 917 


9.56 118 


9.59 205 


0.40 795 


9.96 912 


22 


9.56 150 


9.59 243 


0.40 757 


9.96 907 


38 


23 


9.56 182 


9.59 280 


0.40 720 


9.96 903 


37 


24 


9.56 215 


9.59 317 


0.40 683 


9.96 898 


36 


25 


9.56 247 


9.59 354 


0.40 646 


9.96 893 


35 


26 


9.56 279 


9.59 391 


0.40 609 


9.96 888 


34 


27 


9.56 311 


9.59 429 


0.40 571 


9.95 883 


33 


28 


9.56 343 


9.59 466 


0.40 534 


9.96 878 


32 


29 
30 
31 


9.56 375 


9.59 503 


0.40 497 


9.96 873 


31 
30 

29 


9.56 408 


9.59 540 


0.40 460 


9.96 868 


9.56 440 


9.59 577 


0.40 423 


9.96 863 


32 


9.56 472 


9.59 614 


0.40 386 


9.96 858 


28 


33 


9.56 504 


9.59 651 


0.40 349 


9.96 853 


27 


34 


9.56 536 


9.59 688 


0.40 312 


9.96 848 


26 


35 


9.56 568 


9.59 725 


0.40 275 


9.96 843 


25 


36 


9.56 599 


9.59 762 


0.40 238 


9.96 838 


24 


37 


9.56 631 


9.59 799 


0.40 201 


9.96 833 


23 


38 


9.56 663 


9.59 835 


0.40 165 


9.96 828 


22 


39 
40 

41 


9.56 695 


9.59 872 


0.40 128 


9.96 823 


21 
20 

19 


9.56 727 


9.59 909 


0.40 091 


9.96 818 


9.56 759 


9.59 946 


0.40 054 


9.96 813 


42 


9.56 790 


9.59 983 


0.40 017 


9.96 808 


18 


43 


9.56 822 


9.60 019 


0.39 981 


9.96 803 


17 


44 


9.56 854 


9.60 056 


0.39 944 


9.96 798 


16 


45 


9.56 886 


9.60 093 


0.39 907 


9.96 793 


15 


46 


9.56 917 


9.60 130 


0.39 870 


9.96 788 


14 


47 


9.56 949 


9-60 166 


0.39 834 


9.96 783 


13 


48 


9.56 980 


9.60 203 


0.39 797 


9.96 778 


12 


49 
50 

51 


9.57 012 


9.60 240 


0.39 760 


9.96 772 


11 
10 

9 


9.57 044 


9.60 276 


0.39 724 


9.96 767 


9.57 075 


9.60 313 


0.39 687 


9.96 762 


52 


9.57 107 


9.60 349 


0.39 651 


9.96 757 


8 


53 


9.57 138 


9.60 386 


0.39 614 


9.96 752 


7 


54 


9.57 169 


9.60 422 


0.39 578 


9.96 747 


6 


55 


9.57 201 


9.60 459 


0.39 541 


9.96 742 


5 


56 


9.57 232 


9.60 495 


0.39 505 


9.96 737 


4 


57 


9.57 264 


9.60 532 


0.39 468 


9.96 732 


3 


58 


9.57 295 


9.60 568 


0.39 432 


9.96 727 


2 


59 
60 


9.57 326 


9.60 605 


0.39 395 


9.96 722 


1 



9.57 358 


9.60 641 


0.39 359 


9.96 717 




L. Cos. 


L. Cot. 


L. Tan. 


L. Sin. 


/ 






61 


3° 







22° 



49 



1 


L. Sin. 


L. Tan. 


L. Cot. 


L. Cos. 






1 


9.57 358 


9.60 641 


0.39 359 


9.96 717 


60 

59 


9.57 389 


9.60 677 


0.39 323 


9.96 711 


2 


9.57 420 


9.60 714 


0.39 286 


9.96 706 


58 


3 


9.57 451 


9.60 750 


0.39 250 


9.96 701 


57 


4 


9.57 482 


9.60 786 


0.39 214 


9.96 696 


56 


5 


9.57 514 


9.60 823 


0.39 177 


9.96 691 


55 


6 


9.57 545 


9.60 859 


0.39 141 


9.96 686 


54 


7 


9.57 576 


9.60 895 


0.39 105 


9.96 681 


53 


8 


9.57 607 


9.60 931 


0.39 069 


9.96 676 


52 


9 
10 

11 


9.57 638 


9.60 967 


0.39 033 


9.96 670 


51 
50 

49 


9.57 669 


9.61 004 


0.38 996 


9.96 665 


9.57 700 


9.61 040 


0.38 960 


9.96 660 


12 


9.57 731 


9.61 076 


0.38 924 


9.96 655 


48 


13 


9.57 762 


9.61 112 


0.38 888 


9.96 650 


47 


14 


9.57 793 


9.61 148 


0.38 852 


9.96 645 


46 


15 


9.57 824 


9.61 184 


0.38 816 


9.96 640 


45 


16 


9.57 855 


9.61 220 


0.38 780 


9.96 634 


44 


17 


9.57 885 


9.61 256 


0.38 744 


9.96 629 


43 


18 


9.57 916 


9.61 292 


0.38 708 


9.96 624 


42 


19 
20 

21 


9.57 947 


9.61 328 


0.38 672 


9.96 619 


41 
40 

39 


9.57 978 


9.61364 


0.38 636 


9.96 614 


9.58 008 


9.61 400 


0.38 600 


9.96 608 


22 


9.58 039 


9.61 436 


0.38 564 


9.96 603 


38 


23 


9.58 070 


9.61 472 


0.38 528 


9.96 598 


37 


24 


9.58 101 


9.61 508 


0.38 492 


9.96 593 


36 


25 


9.58 131 


9.61544 


0.38 456 


9.96 588 


35 


26 


9.58 162 


9.61 579 


0.38 421 


9.96 582 


34 


27 


9.58 192 


9.61 615 


0.38 385 


9.96 577 


33 


28 


9.58 223 


9.61 651 


0.38 349 


9.96 572 


32 


29 
30 

31 


9.58 253 


9.61 687 


0.38 313 


9.96 567 


31 
30 

29 


9.58 284 


9.61 722 


0.38 278 


9.96 562 


9.58 314 


9.61 758 


0.38 242 


9.96 556 


32 


9.58 345 


9.61 794 


0.38 206 


9.96 551 


28 


33 


9.58 375 


9.61 830 


0.38 170 


9.96 546 


27 


34 


9.58 406 


9.61 865 


0.38 135 


9.96 541 


26 


35 


9.58 436 


9.61 901 


0.38 099 


9.96 535 


25 


36 


9.58 467 


9.61 936 


0.38 064 


9.96 530 


24 


37 


9.58 497 


9.61 972 


0.38 028 


9.96 525 


23 


38 


9.58 527 


9.62 008 


0.37 992 


9.96 520 


22 


39 
40 
41 


9.58 557 


9.62 043 


0.37 957 


9.96 514 


21 
20 

19 


9.58 588 


9.62 079 


0.37 921 


9.96 509 


9.58 618 


9.62 114 


0.37 886 


9.96 504 


42 


9.58 648 


9.62 150 


0.37 850 


9.96 498 


18 


43 


9.58 678 


9.62 185 


0.37 815 


9.96 493 


17 


44 


9.58 709 


9.62 221 


0.37 779 


9.96 488 


16 


45 


9.58 739 


9.62 256 


0.37 744 


9.96 483 


15 


46 


9.58 769 


9.62 292 


0.37 708 


9.96 477 


14 


47 


9.58 799 


9.62 327 


0.37 673 


9.96 472 


13 


48 


9.58 829 


9.62 362 


0.37 638 


9.96 467 


12 


49 
50 

51 


9.58 859 


9.62 398 


0.37 602 


9.96 461 


11 
10 

9 


9.58 889 


9.62 433 


0.37 567 


9.96 456 


9.58 919 


9.62 468 


0.37 532 


9.96 451 


52 


9.58 949 


9.62 504 


0.37 496 


9.96 445 


8 


53 


9.58 979 


9.62 539 


0.37 461 


9.96 440 


7 


54 


9.59 009 


9.62 574 


0.37 426 


9.96 435 


6 


55 


9.59 039 


9.62 609 


0.37 391 


9.96 429 


5 


56 


9.59 069 


9.62 645 


0.37 355 


9.96 424 


4 


57 


9.59 098 


9.62 680 


0.37 320 


9.96 419 


3 


58 


9.59 128 


9.62 715 


0.37 285 


9.96 413 


2 


59 
60 


9.59 158 


9.62 750 


0.37 250 


9.96 408 


1 



9.59 188 


9.62 785 


0.37 215 


9.96 403 




L. Cos. 


L. Cot. 


L. Tan. 


L. Sin. 


t 



65" 



5o 



23' 



t 


L. Sin. 


L. Tan. 


L. Cot. 


L. Cos. 






1 


9.59 188 


9.62 785 


0.37 215 


9.96 403 


60 

59 


9.59 218 


9.62 820 


0.37 180 


9.96 397 


2 


9.59 247 


9.62 855 


0.37 145 


9.96 392 


58 


3 


9.59 277 


9.62 890 


0.37 110 


9.96 387 


57 


4 


9.59 307 


9.62 926 


0.37 074 


9.96 381 


56 


5 


9.59 336 


9.62 961 


0.37 039 


9.96 376 


55 


6 


9.59 366 


9.62 996 


0.37 004 


9.96 370 


54 


7 


9.59 396 


9.63 031 


0.36 969 


9.96 365 


53 


8 


9.59 425 


9.63 066 


0.36 934 


9.96 360 


52 


9 
10 

11 


9.59 455 


9.63 101 


0.36 899 


9.96 354 


51 
50 

49 


9.59 484 


9.63 135 


0.36 865 


9.96 349 


9.59 514 


9.63 170 


0.36 830 


9.96 343 


12 


9.59 543 


9.63 205 


0.36 795 


9.96 338 


48 


13 


9.59 573 


9.63 240 


0.36 760 


9.96 333 


47 


14 


9.59 602 


9.63 275 


0.36 725 


9.96 327 


46 


15 


9.59 632 


9.63 310 


0.36 690 


9.96 322 


45 


16 


9.59 661 


9.63 345 


0.36 655 


9.96 316 


44 


17 


9.59 690 


9.63 379 


0.36 621 


9.96 311 


43 


18 


9.59 720 


9.63 414 


0.36 586 


9.96 305 


42 


19 
20 
21 


9.59 749 


9.63 449 


0.36 551 


9.96 300 


41 
40 
39 


9.59 778 


9.63 484 


0.36 516 


9.96 294 


9.59 808 


9.63 519 


0.36 481 


9.96 289 


22 


9.59 837 


9.63 553 


0.36 447 


9.96 284 


38 


23 


9.59 866 


9.63 588 


0.36 412 


9.96 278 


37 


24 


9.59 895 


9.63 623 


0.36 377 


9.96 273 


36 


25 


9.59 924 


9.63 657 


0.36 343 


9.96 267 


35 


26 


9.59 954 


9.63 692 


0.36 308 


9.96 262 


34 


27 


9.59 983 


9.63 726 


0.36 274 


9.96 256 


33 


28 


9.60 012 


9.63 761 


0.36 239 


9.96 251 


32 


29 
30 

31 


9.60 041 


9.63 796 


0.36 204 


9.96 245 


31 
30 
29 


9.60 070 


9.63 830 


0.36 170 


9.96 240 


9.60 099 


9.63 865 


0.36 135 


9.96 234 


32 


9.60 128 


9.63 899 


0.36 101 


9.96 229 


28 


33 


9.60 157 


9.63 934 


0.36 066 


9.96 223 


27 


34 


9.60 186 


9.63 968 


0.36 032 


9.96 218 


26 


35 


9.60 215 


9.64 003 


0.35 997 


9.96 212 


25 


36 


9.60 244 


9.64 037 


0.35 963 


9.96 207 


24 


37 


9.60 273 


9.64 072 


0.35 928 


9.96 201 


23 


38 


9.«0 302 


9.64 106 


0.35 894 


9.96 196 


22 


39 
40 

41 


9.60 331 


9.64 140 


0.35 860 


9.96 190 


21 
20 

19 


9.60 359 


9.64 175 


0.35 825 


9.96 185 


9.60 388 


9.64 209 


0.35 791 


9.96 179 


42 


9.60 417 


9.64 243 


0.35 757 


9.96 174 


18 


43 


9.60 446 


9.64 278 


0.35 722 


9.96 168 


17 


44 


9.60 474 


9.64 312 


0.35 688 


9.96 162 


16 


45 


9.60 503 


9.64 346 


0.35 654 


9.96 157 


15 


46 


9.60 532 


9.64 381 


0.35 619 


9.96 151 


14 


47 


9.60 561 


9.64 415 


0.35 585 


9.96 146 


13 


48 


9.60 589 


9.64 449 


0.35 551 


9.96 140 


12 


49 
50 

51 


9.60 618 


9.64 483 


0.35 517 


9.96 135 


11 
10 

9 


9.60 646 


9.64 517 


0.35 483 


9.96 129 


9.60 675 


9.64 552 


0.35 448 


9.96 123 


52 


9.60 704 


9.64 586 


0.35 414 


9.96 118 


8 


53 


9.60 732 


9.64 620 


0.35 380 


9.96 112 


7 


54 


9.60 761 


9.64 654 


0.35 346 


9.96 107 


6 


55 


9.60 789 


9.64 688 


0.35 312 


9.96 101 


5 


56 


9.60 818 


9.64 722 


0.35 278 


9.96 095 


4 


57 


9.60 846 


9.64 756 


0.35 244 


9.96 090 


3 


58 


9.60 875 


9.64 790 


0.35 210 


9.96 084 


2 


59 
60 


9.60 903 


9.64 824 


0.35 176 


9.96 079 


1 



9.60 931 


9.64 858 


0.35 142 


9.96 073 




L. Cos. 


L. Cot. 


L. Tan. 


L. Sin. 


' 



66 



24 c 



51 



t 


L. Sin. 


L. Tan. 


L. Cot. 


L. Cos. 






1 


9.60 931 


9.64 858 


0.35 142 


9.96 073 


60 

59 


9.60 960 


9.64 892 


0.35 108 


9.96 067 


2 


9.60 988 


9.64 926 


0.35 074 


9.96 062 


58 


3 


9.61 016 


9.64 960 


0.35 040 


9.96 056 


57 


4 


9.61 045 


9.64 994 


0.35 006 


9.96 050 


56 


5 


9.61 073 


9.65 028 


0.34 972 


9.96 045 


55 


6 


9.61 101 


9.65 062 


0.34 938 


9.96 039 


54 


7 


9.61 129 


9.65 096 


0.34 904 


9.96 034 


53 


8 


9.61 158 


9.65 130 


0.34 870 


9.96 028 


52 


9 
10 

11 


9.61 186 


9.65 164 


0.34 836 


9.96 022 


51 
50 

49 


9.61 214 


9.65 197 


0.34 803 


9.96 017 


9.61 242 


9.65 231 


0.34 769 


9.96 Oil 


12 


9.61 270 


9.65 265 


0.34 735 


9.96 005 


48 


13 


9.61 298 


9.65 299 


0.34 701 


9.96 000 


47 


14 


9.61 326 


9.65 333 


0.34 667 


9.95 994 


46 


15 


9.61 354 


9.65 366 


0.34 634 


9.95 988 


45 


16. 


9.61 382 


9.65 400 


0.34 600 


9.95 982 


44 


17 


9.61 411 


9.65 434 


0.34 566 


9.95 977 


43 


18 


9.61 438 


9.65 467 


0.34 533 


9.95 971 


42 


19 
20 

21 


9.61 466 


9.65 501 


0.34 499 


9.95 965 


41 
40 

39 


9.61 494 


9.65 535 


0.34 465 


9.95 960 


9.61 522 


9.65 568 


0.34 432 


9.95 954 


22 


9.61 550 


9.65 602 


0.34 398 


9.95 948 


38 


23 


9.61 578 


9.65 636 


0.34 364 


9.95 942 


37 


24 


9.61 606 


9.65 669 


0.34 331 


9.95 937 


36 


25 


9.61 634 


9.65 703 


0.34 297 


9.95 931 


35 


26 


9.61 662 


9.65 736 


0.34 264 


9.95 925 


' 34 


27 


9.61 689 


9.65 770 


0.34 230 


9.95 920 


33 


28 


9.61 717 


9.65 803 


0.34 197 


9.95 914 


32 


29 
30 

31 


9.61 745 


9.65 837 


0.34 163 


9.95 908 


31 
30 

29 


9.61 773 


9.65 870 


0.34 130 


9.95 902 


9.61 800 


9.65 904 


0.34 096 


9.95 897 


32 


9.61 828 


9.65 937 


0.34 063 


9.95 891 


28 


33 


9.61 856 


9.65 971 


0.34 029 


9.95 885 


27 


34 


9.61 883 


9.66 004 


0.33 996 


9.95 879 


26 


35 


9.61 911 


9.66 038 


0.33 962 


9.95 873 


25 


36' 


9.61 939 


9.66 071 


0.33 929 


9.95 868 


24 


37 


9.61 966 


9.66 104 


0.33 896 


9.95 862 


23 


38 


9.61 994 


9.66 138 


0.33 862 


9.95 856 


22 


39 
40 
41 


9.62 021 


9.66 171 


0.33 829 


9.95 850 


21 
20 

19 


9.62 049 


9.66 204 


0.33 796 


9.95 844 


9.62 076 


9.66 238 


0.33 762 


9.95 839 


42 


9.62 104 


9.66 271 


0.33 729 


9.95 833 


18 


43 


9.62 131 


9.66 304 


0.33 696 


9.95 827 


17 


44 


9.62 159 


9.66 337 


0.33 663 


9.95 821 


16 


45 


9.62 186 


9.66 371 


0.33 629 


9.95 815 


15 


46 


9.62 214 


9.66 404 


0.33 596 


9.95 810 


14 


47 


9.62 241 


9.66 437 


0.33 563 


9.95 804 


13 


48 


9.62 268 


9.66 470 


0.33 530 


9.95 798 


12 


49 
50 
51 


9.62 296 


9.66 503 


0.33 497 


9.95 792 


11 
10 

9 


9.62 323 


9.66 537 


0.33 463 


9.95 786 


9.62 350 


9.66 570 


0.33 430 


9.95 780 


52 


9.62 377 


9.66 603 


0.33 397 


9.95 775 


8 


53 


9.62 405 


9.66 636 


0.33 364 


9.95 769 


7 


54 


9.62 432 


9.66 669 


0.33 331 


9.95 763 


6 


55 


9.62 459 


9.66 702 


0.33 298 


9.95 757 


5 


56 


9.62 486 


9.66 735 


0.33 265 


9.95 751 


4 


57 


9.62 513 


9.66 768 


0.33 232 


9.95 745 


3 


58 


9.62 541 


9.66 801 


0.33 199 


9.95 739 


2 


59 
60 


9.62 568 


9.66 834 


0.33 166 


9.95 733 


1 



9.62 595 


9.66 867 


0.33 133 


9.95 728 




L. Cos. 


L. Cot. 


L. Tan. 


L. Sin. 


t 



65 



52 



25' 



/ 


L. Sin. 


L. Tan. 


L. Cot. 


L. Cos. 






1 


9.62 595 


9.66 867 


0.33 133 


9.95 728 


60 

59 


9.62 622 


9.66 900 


0.33 100 


9.95 722 


2 


9.62 649 


9.66 933 


0.33 067 


9.95 716 


58 


3 


9.62 676 


9.66 966 


0.33 034 


9.95 710 


57 


4 


9.62 703 


9.66 999 


0.33 001 


9.95 704 


56 


5 


9.62 730 


9.67 032 


0.32 968 


9.95 698 


55 


6 


9.62 757 


9.67 065 


0.32 935 


9.95 692 


54 


7 


9.62 784 


9.67 098 


0.32 902 


9.95 686 


53 


8 


9.62 811 


9.67 131 


0.32 869 


9.95 680 


52 


9 
10 

11 


9.62 838 


9.67 163 


0.32 837 


9.95 674 


51 
50 

49 


9.62 865 


9.67 196 


0.32 804 


9.95 668 


9.62 892 


9.67 229 


0.32 771 


9.95 663 


12 


9.62 918 


9.67 262 


0.32 738 


9.95 657 


48 


13 


9.62 945 


9.67 295 


0.32 705 


9.95 651 


47 


14 


9.62 972 


9.67 327 


0.32 673 


9.95 645 


46 


15 


9.62 999 


9.67 360 


0.32 640 


9.95 639 


45 


16 


9.63 026 


9.67 393 


0.32 607 


9.95 633 


44 


17 


9.63 052 


9.67 426 


0.32 574 


9.95 627 


43 


18 


9.63 079 


9.67 458 


0.32 542 


9.95 621 


42 


19 
20 

21 


9.63 106 


9.67 491 


0.32 509 


9.95 615 


41 
40 

39 


9.63 133 


9.67 524 


0.32 476 


9.95 609 


9.63 159 


9.67 556 


0.32 444 


9.95 603 


22 


9.63 186 


9.67 589 


0.32 411 


9.95 597 


38 


23 


9.63 213 


9.67 622 


0.32 378 


9.95 591 


37 


24 


9.63 239 


9.67 654 


0.32 346 


9.95 585 


36 


25 


9.63 266 


9.67 687 


0.32 313 


9.95 579 


35 


26 


9.63 292 


9.67 719 


0.32 281 


9.95 573 


34 


27 


9.63 319 


9.67 752 


0.32 248 


9.95 567 


33 


28 


9.63 345 


9.67 785 


0.32 215 


9.95 561 


32 


29 
30 

31 


9.63 372 


9.67 817 


0.32 183 


9.95 555 


31 
30 

29 


9.63 398 


9.67 850 


0.32 150 


9.95 549 


9.63 425 


9.67 882 


0.32 118 


9.95 543 


32 


9.63 451 


9.67 915 


0.32 085 


9.95 537 


28 


33 


9.63 478 


9.67 947 


0.32 053 


9.95 531 


27 


34 


9.63 504 


9.67 980 


0.32 020 


9.95 525 


26 


35 


9.63 531 


9.68 012 


0.31 988 


9.95 519 


25 


36 


9.63 557 


9.68 044 


0.31 956 


9.95 513 


24 


37 


9.63 583 


9.68 077 


0.31 923 


9.95 507 


23 


38 


9.63 610 


9.68 109 


0.31 891 


9.95 500 


22 


39 
40 
41 


9.63 636 


9.68 142 


0.31 858 


9.95 494 


21 
20 

19 


9.63 662 


9.(58 174 


0.31 826 


9.95 488 


9.63 689 


9.68 206 


0.31 794 


9.95 482 


42 


9.63 715 


9.68 239 


0.31 761 


9.95 476 


18 


43 


9.63 741 


9.68 271 


0.31 729 


9.95 470 


17 


44 


9.63 767 


9.68 303 


0.31 697 


9.95 464 


16 


45 


9.63 794 


9.68 336 


0.31 664 


9.95 458 


15 


46 


9.63 820 


9.68 368 


0.31 632 


9.95 452 


14 


47 


9.63 846 


9.68 400 


0.31 600 


9.95 446 


13 


48 


9.63 872 


9.68 432 


0.31 568 


9.95 440 


12 


49 
50 

51 


9.63 898 


9.68 465 


0.31 535 


9.95 434 


11 
10 

9 


9.63 924 


9.68 497 


0.31 503 


9.95 427 


9.63 950 


9.68 529 


0.31 471 


9.95 421 


52 


9.63 976 


9.68 561 


0.31 439 


9.95 415 


8 


53 


9.64 002 


9.68 593 


0.31 407 


9.95 409 


7 


54 


9.64 028 


9.68 626 


0.31 374 


9.95 403 


6 


55 


9.64 054 


9.68 658 


0.31 342 


9.95 397 


5 


56 


9.64 080 


9.68 690 


0.31 310 


9.95 391 


4 


57 


9.64 106 


9.68 722 


0.31 278 


9.95 384 


3 


58 


9.64 132 


9.68 754 


0.31 246 


9.95 378 


2 


59 
60 


9.64 158 


9.68 786 


0.31 214 


9.95 372 


1 



9.64 184 


9.68 818 


0.31 182 


9.95 366 




L. Cos. 


L. Cot. 


L. Tan. 


L. Sin. 


/ 



64' 



26' 



53 



t 


L. Sin. 


L. Tan. 


L. Cot. 


L. Cos. 






1 


9.64 184 


9.68 818 


0.31 182 


9.95 366 


60 

59 


9.64 210 


9.68 850 


0.31 150 


9.95 360 


2 


9.64 236 


9.68 882 


0.31 118 


9.95 354 


58 


3 


9.64 262 


9.68 914 


0.31 086 


9.95 348 


57 


4 


9.64 288 


9.68 946 


0.31 054 


9.95 341 


56 


5 


9.64 313 


9.68 978 


0.31 022 


9.95 335 


55 


6 


9.64 339 


9.69 010 


0.30 990 


9.95 329 


54 


7 


9.64 365 


9.69 042 


0.30 958 


9.95 323 


53 


8 


9.64 391 


9.69 074 


0.30 926 


9.95 317 


52 


9 
10 

11 


9.64 417 


9.69 106 


0.30 894 


9.95 310 


51 
50 

49 


9.64 442 


9.69 138 


0.30 862 


9.95 304 


9.64 468 


9.69 170 


0.30 830 


9.95 298 


12 


9.64 494 


9.69 202 


0.30 798 


9.95 292 


48 


13 


9.64 519 


9.69 234 


0.30 766 


9.95 286 


47 


14 


9.64 545 


9.69 266 


0.30 734 


9.95 279 


46 


15 


9.64 571 


9.69 298 


0.30 702 


9.95 273 


45 


16 


9.64 596 


9.69 329 


0.30 671 


9.95 267 


44 


17 


9.64 622 


9.69 361 


0.30 639 


9.95 261 


43 


18 


9.64 647 


9.69 393 


0.30 607 


9.95 254 


42 


19 
20 
21 


9.64 673., 


9.69 425 


0.30 575 


9.95 248 


41 
40 

39 


9.64 698 


9.69 457 


0.30 543 


9.95 242 


9.64 724 


9.69 488 


0.30 512 


9.95 236 


22 


9.64 749 


9.69 520 


0.30 480 


9.95 229 


38 


23 


9.64 775 


9.69 552 


0.30 448 


9.95 223 


37 


24 


9.64 800 


9.69 584 


0.30 416 


9.95 217 


36 


25 


9.64 826 


9.69 615 


0.30 385 


9.95 211 


35 


26 


9.64 851 


9.69 647 


0.30 353 


9.95 204 


34 


27 


9.64 877 


9.69 679 


0.30 321 


9.95 198 


33 


28 


9.64 902 


9.69 710 


0.30 290 


9.95 192 


32 


29 
30 

31 


9.64 927 


9.69 742 


0.30 258 


9.95 185 


31 
30 

29 


9.64 953 


9.69 774 


0.30 226 


9.95 179 


9.64 978 


9.69 805 


0.30 195 


9.95 173 


32 


9.65 003 


9.69 837 


0.30 163 


9.95 167 


28 


33 


9.65 029 


9.69 868 


0.30 132 


9.95 160 


27 


34 


9.65 054 


9.69 900 


0.30 100 


9.95 154 


26 


35 


9.65 079 


9.69 932 


0.30 068 


9.95 148 


25 


36 


9.65 104 


9.69 963 


0.30 037 


9.95 141 


24 


37 


9.65 130 


9.69 995 


0.30 005 


9.95 135 


23 


38 


9.65 155 


9.70 026 


0.29 974 


9.95 129 


22 


39 
40 
41 


9.65 180 


9.70 058 


0.29 942 


9.95 122 


21 
20 

19 


9.65 205 


9.70 089 


0.29 911 


9.95 116 


9.65 230 


9.70 121 


0.29 879 


9.95 110 


42 


9.65 255 


9.70 152 


0.29 848 


9.95 103 


18 


43 


9.65 281 


9.70 184 


0.29 816 


9.95 097 


17 


44 


9.65 306 


9.70 215 


0.29 785 


9.95 090 


16 


45 


9.65 331 


9.70 247 


0.29 753 


9.95 084 


15 


46 


9.65 356 


9.70 278 


0.29 722 


9.95 078 


14 


47 


9.65 381 


9.70 309 


0.29 691 


9.95 071 


13 


48 


9.65 406 


9.70 341 


0.29 659 


9.95 065 


12 


49 
50 

51 


9.65 431 


9.70 372 


0.29 628 


9.95 059 


11 
10 

9 


9.65 456 


9.70 404 


0.29 596 


9.95 052 


9.65 481 


9.70 435 


0.29 565 


9.95 046 


52 


9.65 506 


9.70 466 


0.29 534 


9.95 039 


8 


53 


9.65 531 


9.70 498 


0.29 502 


9.95 033 


7 


54 


9.65 556 


9.70 529 


0.29 471 


9.95 027 


6 


55 


9.65 580 


9.70 560 


0.29 440 


9.95 020 


5 


56 


9.65 605 


9.70 592 


0.29 408 


9.95 014 


4 


57 


9.65 630 


9.70 623 


0.29 377 


9.95 007 


3 


58 


9.65 655 


9.70 654 


0.29 346 


9.95 001 


2 


59 
60 


9.65 680 


9.70 685 


0.29 315 


9.94 995 


1 



9.65 705 


9.70 717 


0.29 283 


9.94 988 




L. Cos. 


L. Cot. 


L. Tan. 


L. Sin. 


' 



63 



54 



27° 



t 


L. Sin. 


L. Tan. 


L. Cot. 


L. Cos. 






1 


9.65 705 


9.70 717 


0.29 283 


9.94 988 


60 

59 


9.65 729 


9.70 748 


0.29 252 


9.94 982 


2 


9.65 754 


9.70 779 


0.29 221 


9.94 975 


58 


3 


9.65 779 


9.70 810 


0.29 190 


9.94 969 


57 


4 


9.65 804 


9.70 841 


0.29 159 


9.94 962 


56 


5 


9.65 828 


9.70 873 


0.29 127 


9.94 956 


55 


6 


9.65 853 


9.70 904 


0.29 096 


9.94 949 


54 


7 


9.65 878 


9.70 935 


0.29 065 


9.94 943 


53 


8 


9.65 902 


9.70 966 


0.29 034 


9.94 936 


52 


9 
10 

11 


9.65 927 


9.70 997 


0.29 003 


9.94 930 


51 
50 

49 


9.65 952 


9.71 028 


0.28 972 


9.94 923 


9.65 976 


9.71 059 


0.28 941 


9.94 917 


12 


9.66 001 


9.71 090 


0.28 910 


9.94 911 


48 


13 


9.66 025 


9.71 121 


0.28 879 


9.94 904 


47 


14 


9.66 050 


9.71 153 


0.28 847 


9.94 898 


46 


15 


9.66 075 


9.71 184 


0.28 816 


9.94 891 


45 


16 


9.66 099 


9.71 215 


0.28 785 


9.94 885 


44 


17 


9.66 124 


9.71 246 


0.28 754 


9.94 878 


43 


18 


9.66 148 


9.71 277 


0.28 723 


9.94 871 


42 


19 
20 

21 


9.66 173 


9.71 308 


0.28 692 


9.94 865 


41 
40 

39 


9.66 197 


9.71 339 


0.28 661 


9.94 858 


9.66 221 


9.71 370 


0.28 630 


9.94 852 


22 


9.66 216 


9.71 401 


0.28 599 


9.94 845 


38 


23 


9.66 270 


9.71 431 


0.28 569 


9.94 839 


37 


24 


9.66 295 


9.71 462 


0.28 538 


9.94 832 


36 


25 


9.66 319 


9.71 493 


0.28 507 


9.94 826 


35 


26 


9.66 343 


9.71 524 


0.28 476 


9.94 819 


34 


27 


9.66 368 


9.71 555 


0.28 445 


9.94 813 


33 


28 


9.66 392 


9.71 586 


0.28 414 


9.94 806 


32 


29 
30 
31 


9.66 416 


9.71 617 


0.28 383 


9.94 799 


31 
30 

29 


9.66 441 


9.71 648 


0.28 352 


9.94 793 


9.66 465 


9.71 679 


0.28 321 


9.94 786 


32 


9.66 489 


9.71 709 


0.28 291 


9.94 780 


28 


33 


9.66 513 


9.71 740 


0.28 260 


9.94 773 


27 


34 


9.66 537 


9.71 771 


0.28 229 


9.94 767 


26 


35 


9.66 562 


9.71 802 


0.28 198 


9.94 760 


25 


36 


9.66 586 


9.71 833 


0.28 167 


9.94 753 


24 


37 


9.66 610 


9.71 863 


0.28 137 


9.94 747 


23 


38 


9.66 634 


9.71 894 


0.28 106 


9.94 740 


22 


39 
40 
41 


9.66 658 


9.71 925 


0.28 075 


9.94 734 


21 
20 
19 


9.66 682 


9.71 955 


0.28 045 


9.94 727 


9.66 706 


9.71 986 


0.28 014 


9.94 720 


42 


9.66 731 


9.72 017 


0.27 983 


9.94 714 


18 


43 


9.66 755 


9.72 048 


0.27 952 


9.94 707 


17 


44 


9.66 779 


9.72 078 


0.27 922 


9.94 700 


16 


45 


9.66 803 


9.72 109 


0.27 891 


9.94 694 


15 


46 


9.66 827 


9.72 140 


0.27 860 


9.94 687 


14 


47 


9.66 851 


9.72 170 


0.27 830 


9.94 680 


13 


48 


9.66 875 


9.72 201 


0.27 799 


9.94 674 


12 


49 
50 

51 


9.66 899 


9.72 231 


0.27 769 


9.94 667 


11 
10 

9 


9.66 922 


9.72 262 


0.27 738 


9.94 660 


9.66 946 


9.72 293 


0.27 707 


9.94 654 


52 


9.66 970 


9.72 323 


0.27 677 


9.94 647 


8 


53 


9.66 994 


9.72 354 


0.27 646 


9.94 640 


7 


54 


9.67 018 


9.72 384 


0.27 616 


9.94 634 


6 


55 


9.67 042 


9.72 415 


0.27 585 


9.94 627 


5 


56 


9.67 066 


9.72 445 


0.27 555 


9.94 620 


4 


57 


9.67 090 


9.72 476 


0.27 524 


9.94 614 


3 


58 


9.67 113 


9.72 506 


0.27 494 


9.94 607 


2 


59 
60 


9.67 137 


9.72 537 


0.27 463 


9.94 600 


1 



9.67 161 


9.72 567 


0.27 433 


9.94 593 




L. Cos. 


L. Cot. 


L. Tan. 


L. Sin. 


t 



62' 



^M 



■■ 



28' 



55 



t 


L. Sin. 


L. Tan. 


L. Cot. 


L. Cos. 






1 


9.67 161 


9.72 567 


0.27 433 


9.94 593 


60 

59 


9.67 185 


9.72 598 


0.27 402 


9.94 587 


2 


9.67 208 


9.72 628 


0.27 372 


9.94 580 


58 


3 


9.67 232 


9.72 659 


0.27 341 


9.94 573 


57 


4 


9.67 256 


9.72 689 


0.27 311 


9.94 567 


56 


5 


9.67 280 


9.72 720 


0.27 280 


9.94 560 


55 


6 


9.67 303 


9.72 750 


0.27 250 


9.94 553 


54 


7 


9.67 327 


9.72 780 


0.27 220 


9.94 546 


53 


8 


9.67 350 


9.72 811 


0.27 189 


9.94 540 


52 


9 
10 

11 


9.67 374 


9.72 841 


0.27 159 


9.94 533 


51 
50 

49 


9.67 398 


9.72 872 


0.27 128 


9.94 526 


9.67 421 


9.72 902 


0.27 098 


9.94 519 


12 


9.67 445 


9.72 932 


0.27 068 


9.94 513 


48 


13 


9.67 468 


9.72 963 


0.27 037 


9.94 506 


47 


14 


9.67 492 


9.72 993 


0.27 007 


9.94 499 


46 


15 


9.67 515 


9.73 023 


0.26 977 


9.94 492 


45 


16 


9.67 539 


9.73 054 


0.26 946 


9.94 485 


44 


17 


9.67 562 


9.73 084 


0.26 916 


9.94 479 


43 


18 


9.67 586 


9.73 114 


0.26 886 


9.94 472 


42 


19 
20 
21 


9.67 609 


9.73 144 


0.26 856 


9.94 465 


41 
40 

39 


9.67 633 


9.73 175 


0.26 825 


9.94 458 


9.67 656 


9.73 205 


0.26 795 


9.94 451 


22 


9.67 680 


9.73 235 


0.26 765 


9.94 445 


38 


23 


9.67 703 


9.73 265 


0.26 735 


9.94 438 


37 


24 


9.67 726 


9.73 295 


0.26 705 


9.94 431 


36 


25 


9.67 750 


9.73 326 


0.26 674 


9.94 424 


35 


26 


9.67 773 


9.73 356 


0.26 644 


9.94 417 


34 


27 


9.67 796 


9.73 386 


0.26 614 


9.94 410 


33 


28 


9.67 820 


9.73 416 


0.26 584 


9.94 404 


32 


29 
30 

31 


9.67 843 


9.73 446 


0.26 554 


9.94 397 


31 
30 

29 


9.67 866 


9.73 476 


0.26 524 


9.94 390 


9.67 890 


9.73 507 


0.26 493 


9.94 383 


32 


9.67 913 


9.73 537 


0.26 463 


9.94 376 


28 


33 


9.67 936 


9.73 567 


0.26 433 


9.94 369 


27 


34 


9.67 959 


9.73 597 


0.26 403 


9.94 362 


26 


35 


9.67 982 


9.73 627 


0.26 373 


9.94 355 


25 


36 


9.68 006 


9.73 657 


0.26 343 


9.94 349 


24 


37 


9.68 029 


9.73 687 


0.26 313 


9.94 342 


23 


38 


9.68 052 


9.73 717 


0.26 283 


9.94 335 


22 


39 
40 
41 


9.68 075 


9.73 747 


0.26 253 


9.94 328 


21 
20 

19 


9.68 098 


9.73 777 


0.26 223 


9.94 321 


9.68 121 


9.73 807 


0.26 193 


9.94 314 


42 


9.68 144 


9.73 837 


0.26 163 


9.94 307 


18 


43 


9.68 167 


9.73 867 


0.26 133 


9.94 300 


17 


44 


9.68 190 


9.73 897 


0.26 103 


9.94 293 


16 


45 


9.68 213 


9.73 927 


0.26 073 


9.94 286 


15 


46 


9.68 237 


9.73 957 


0.26 043 


9.94 279 


14 


47 


9.68 260 


9.73 987 


0.26 013 


9.94 273 


13 


48 


9.68 283 


9.74 017 


0.25 983 


9.94 266 


12 


49 
50 
51 


9.68 305 


9.74 047 


0.25 953 


9.94 259 


11 
10 

9 


9.68 328 


9.74 077 


0.25 923 


9.94 252 


9.68 351 


9.74 107 


0.25 893 


9.94 245 


52 


9.68 374 


9.74 137 


0.25 863 


9.94 238 


8 


53 


9.68 397 


9.74 166 


0.25 834 


9.94 231 


7 


54 


9.68 420 


9.74 196 


0.25 804 - 


9.94 224 


6 


55 


9.68 443 


9.74 226 


0.25 774 


9.94 217 


5 


56 


9.68 466 


9.74 256 


0.25 744 


9.94 210 


4 


57 


9.68 489 


9.74 286 


0.25 714 


9.94 203 


3 


58 


9.68 512 


9.74 316 


0.25 684 


9.94 196 


2 


59 
60 


9.68 534 


9.74 345 


0.25 655 


9.94 189 


1 



9.68 557 


9.74 375 


0.25 625 


9.94 182 




L. Cos. 


L. Cot. 


L. Tan. 


L. Sin. 


t 



61 



56 



29' 



t 


L. Sin. 


L. Tan. 


L. Cot. 


L. Cos. 






1 


9.68 557 


9.74 375 


0.25 625 


9.94 182 


60 

59 


9.68 580 


9.74 405 


0.25 595 


9.94 175 


2 


9.68 603 


9.74 435 


0.25 565 


9.94 168 


58 


3 


9.68 625 


9.74 465 


0.25 535 


9.94 161 


57 


4 


9.68 648 


9.74 494 


0.25 506 


9.94 154 


56 


5 


9.68 671 


9.74 524 


0.25 476 


9.94 147 


55 


6 


9.68 694 


9.74 554 


0.25 446 


9.94 140 


54 


7 


9.68 716 


9.74 583 


0.25 417 


9.94 133 


53 


8 


9.68 739 


9.74 613 


0.25 387 


9.94 126 


52 


9 
10 

11 


9.68 762 


9.74 643 


0.25 357 


9.94 119 


51 
50 
49 


9.68 784 


9.74 673 


0.25 327 


9.94 112 


9.68 807 


9.74 702 


0.25 298 


9.94 105 


12 


9.68 829 


9.74 732 


0.25 268 


9.94 098 


48 


13 


9.68 852 


9.74 762 


0.25 238 


9.94 090 


47 


14 


9.68 875 


9.74 791 


0.25 209 


9.94 083 


46 


15 


9.68 897 


9.74 821 


0.25 179 


9.94 076 


45 


16 


9.68 920 


9.74 851 


0.25 149 


9.94 069 


44 


17 


9.68 942 


9.74 880 


0.25 120 


9.94 062 


43 


18 


9.68 965 


9.74 910 


0.25 090 


9.94 055 


42 


19 
20 
21 


9.68 987 


9.74 939 


0.25 061 


9.94 048 


41 
40 
39 


9.69 010 


9.74 969 


0.25 031 


9.94 041 


9.69 032 


9.74 998 


0.25 002 


9.94 034 


22 


9.69 055 


9.75 028 


0.24 972 


9.94 027 


38 


23 


9.69 077 


9.75 058 


0.24 942 


9.94 020 


37 


24 


9.69 100 


9.75 087 


0.24 913 


9.94 012 


36 


25 


9.69 122 


9.75 117 


0.24 883 


9.94 005 


35 


26 


9.69 144 


9.75 146 


0.24 854 


9.93 998 


34 


27 


9.69 167 


9.75 176 


0.24 824 


9.93 991 


33 


28 


9.69 189 


9.75 205 


0.24 795 


9.93 984 


32 


29 
30 

31 


9.69 212 


9.75 235 


0.24 765 


9.93 977 


31 
30 
29 


9.69 234 


9.75 264 


0.24 736 


9.93 970 


9.69 256 


9.75 294 


0.24 706 


9.93 963 


32 


9.69 279 


9.75 323 


0.24 677 


9.93 955 


28 


33 


9.69 301 


9.75 353 


0.24 647 


9.93 948 


27 


34 


9.69 323 


9.75 382 


0.24 618 


9.93 941 


26 


35 


9.69 345 


9.75 411 


0.24 589 


9.93 934 


25 


36 


9.69 368 


9.75 441 


0.24 559 


9.93 927 


24 


37 


9.69 390 


9.75 470 


0.24 530 


9.93 920 


23 


38 


9.69 412 


9.75 500 


0.24 500 


9.93 912 


22 


39 
40 

41 


9.69 434 


9.75 529 


0.24 471 


9.93 905 


21 
20 

19 


9.69 456 


9.75 558 


0.24 442 


9.93 898 


9.69 479 


9.75 588 


0.24 412 


9.93 891 


42 


9.69 501 


9.75 617 


0.24 383 


9.93 884 


18 


43 


9.69 523 


9.75 647 


0.24 353 


9.93 876 


17 


44 


9.69 545 


9.75 676 


0.24 324 


9.93 869 


16 


45 


9.69 567 


9.75 705 


0.24 295 


9.93 862 


15 


46 


9.69 589 


9.75 735 


0.24 265 


9.93 855 


14 


47 


9.69 611 


9.75 764 


0.24 236 


9.93 847 


13 


48 


9.69 633 


9.75 793 


0.24 207 


9.93 840 


12 


49 
50 

51 


9.69 655 


9.75 822 


0.24 178 


9.93 833 


11 
10 

9 


9.69 677 


9.75 852 


0.24 148 


9.93 826 


9.69 699 


9.75 881 


0.24 119 


9.93 819 


52 


9.69 721 


9.75 910 


0.24 090 


9.93 811 


8 


53 


9.69 743 


9.75 939 


0.24 061 


9.93 804 


7 


54 


9.69 765 


9.75 969 


0.24 031 


9.93 797 


6 


55 


9.69 787 


9.75 998 


0.24 002 


9.93 789 


5 


56 


9.69 809 


9.76 027 


0.23 973 


9.93 782 


4 


57 


9.69 831 


9.76 056 


0.23 944 


9.93 775 


3 


58 


9.69 853 


9.76 086 


0.23 914 


9.93 768 


2 


59 
60 


9.69 875 


9.76 115 


0.23 885 


9.93 760 


1 




9.69 897 


9.76 144 


0.23 856 


9.93 753 




L. COS. 


L. Cot. 


L. Tan. 


L. Sin. 


' 1 






6 


0° 







30' 



57 



/ 


L. Sin. 


L. Tan. 


L. Cot. 


L. Cos. 






1 


9.69 897 


9.76 144 


0.23 856 


9.93 753 


60 

59 


9.69 919 


9.76 173 


0.23 827 


9.93 746 


2 


9.69 941 


9.76 202 


0.23 798 


9.93 738 


58 


3 


9.69 963 


9.76 231 


0.23 769 


9.93 731 


57 


4 


9.69 984 


9.76 261 


0.23 739 


9.93 724 


56 


5 


9.70 006 


9.76 290 


0.23 710 


9.93 717 


55 


6 


9.70 028 


9.76 319 


0.23 681 


9.93 709 


54 


7 


9.70 050 


9.76 348 


0.23 652 


9.93 702 


53 


8 


9.70 072 


9.76 377 


0.23 623 


9.93 695 


52 


9 
10 

11 


9.70 093 


9.76 406 


0.23 594 


9.93 687 


51 
50 

49 


9.70 115 


9.76 435 


0.23 565 


9.93 680 


9.70 137 


9.76 464 


0.23 536 


9.93 673 


12 


9.70 159 


9.76 493 


0.23 507 


9.93 665 


48 


13 


9.70 180 


9.76 522 


0.23 478 


9.93 658 


47 


14 


9.70 202 


9.76 551 


0.23 449 


9.93 650 


46 


15 


9.70 224 


9.76 580 


0.23 420 


9.93 643 


45 


16 


9.70 245 


9.76 609 


0.23 391 


9.93 636 


44 


17 


9.70 267 


9.76 639 


0.23 361 


9.93 628 


43 


18 


9.70 288 


9.76 668 


0.23 332 


9.93 621 


42 


19 
20 

21 


9.70 310 


9.76 697 


0.23 303 


9.93 614 


41 
40 

39 


9.70 332 


9.76 725 


0.23 275 


9.93 606 


9.70 353 


9.76 754 


0.23 246 


9.93 599 


22 


9.70 375 


9.76 783 


0.23 217 


9.93 591 


38 


23 


9.70 396 


9.76 812 


0.23 188 


9.93 584 


37 


24 


9.70 418 


9.76 841 


0.23 159 


9.93 577 


36 


25 


9.70 439 


9.76 870 


0.23 130 


9.93 569 


35 


26 


9.70 461 


9.76 899 


0.23 101 


9.93 562 


34 


27 


9.70 482 


9.76 928 


0.23 072 


9.93 554 


33 


28 


9.70 504 


9.76 957 


0.23 043 


9.93 547 


32 


29 
30 
31 


9.70 525 


9.76 986 


0.23 014 


9.93 539 


31 
30 

29 


9.70 547 


9.77 015 


0.22 985 


9.93 532 


9.70 568 


9.77 044 


0.22 956 


9.93 525 


32 


9.70 590 


9.77 073 


0.22 927 


9 93 517 


28 


33 


9.70 611 


9.77 101 


0.22 899 


9.93 510 


27 


34 


9.70 633 


9.77 130 


0.22 870 


9.93 502 


26 


35 


9.70 654 


9.77 159 


0.22 841 


9.93 495 


25 


36 


9.70 675 


9.77 188 


0.22 812 


9.93 487 


24 


37 


9.70 697 


9.77 217 


0.22 783 


9.93 480 


23 


38 


9.70 718 


9.77 246 


0.22 754 


9.93 472 


22 


39 
40 
41 


9.70 739 


9.77 274 


0.22 726 


9.93 465 


21 
20 

19 


9.70 761 


9.77 303 


0.22 697 


9.93 457 


9.70 782 


9.77 332 


0.22 668 


9.93 450 


42 


9.70 803 


9.77 361 


0.22 639 


9.93 442 


18 


43 


9.70 824 


9.77 390 


0.22 610 


9.93 435 


17 


44 


9.70 846 


9.77 418 


0.22 582 


9.93 427 


16 


45 


9.70 867 


9.77 447 


0.22 553 


9.93 420 


15 


46 


9.70 888 


9.77 476 


0.22 524 


9.93 412 


14 


47 


9.70 909 


9.77 505 


0.22 495 


9.93 405 


13 


48 


9.70 931 


9.77 533 


0.22 467 


9.93 397 


12 


49 
50 

51 


9.70 952 


9.77 562 


0.22 438 


9.93 390 


11 
10 

9 


9.70 973 


9.77 591 


0.22 409 


9.93 382 


9.70 994 


9.77 619 


0.22 381 


9.93 375 


52 


9.71 015 


9.77 648 


0.22 352 


9.93 367 


8 


53 


9.71 036 


9.77 677 


0.22 323 


9.93 360 


7 


54 


9.71 058 


9.77 706 


0.22 294 


9.93 352 


6 


55 


9.71 079 


9.77 734 


0.22 266 


9.93 344 


5 


56 


9.71 100 


9.77 763 


0.22 237 


9.93 337 


4 


57 


9.71 121 


9.77 791 


0.22 209 


9.93 329 


3 


58 


9.71 142 


9.77 820 


0.22 180 


9.93 322 


2 


59 
60 


9.71 163 


9.77 849 


0.22 151 


9.93 314 


1 



9.71 184 


9.77 877 


0.22 123 


9.93 307 




L. Cos. 


L. Cot. 


L. Tan. 


| L. Sin. 


/ 



59 



58 



31 c 



f 


L. Sin. 


L. Tan. 


L. Cot. 


L. Cos. 






1 


9.71 184 


9.77 877 


0.22 123 


9.93 307 


60 

59 


9.71 205 


9.77 906 


0.22 094 


9.93 299 


2 


9.71 226 


9.77 935 


0.22 065 


9.93 291 


58 


3 


9.71 247 


9.77 963 


0.22 037 


9.93 284 


57 


4 


9.71 268 


9.77 992 


0.22 008 


9.93 276 


56 


5 


9.71 289 


9.78 020 


0.21 980 


9.93 269 


55 


6 


9.71 310 


9.78 049 


0.21 951 


9.93 261 


54 


7 


9.71 331 


9.78 077 


0.21 923 


9.93 253 


53 


8 


9.71 352 


9.78 106 


0.21 894 


9.93 246 


52 


9 
10 

11 


9.71 373 


9.78 135 


0.21 865 


9.93 238 


51 
50 

49 


9.71 393 


9.78 163 


0.21 837 


9.93 230 


9.71 414 


9.78 192 


0.21 808 


9.93 223 


12 


9.71 435 


9.78 220 


0.21 780 


9.93 215 


48 


13 


9.71 456 


9.78 249 


0.21 751 


9.93 207 


47 


14 


9.71 477 


9.78 277 


0.21 723 


9.93 200 


46 


15 


9.71 498 


9.78 306 


0.21 694 


9.93 192 


45 


16 


9.71 519 


9.78 334 


0.21 666 


9.93 184 


44 


17 


9.71 539 


9.78 363 


0.21 637 


9.93 177 


43 


18 


9.71 560 


9.78 391 


0.21 609 


9.93 169 


42 


19 
20 
21 


9.71 581 


9.78 419 


0.21 581 


9.93 161 


41 
40 

39 


9.71 602 


9.78 448 


0.21 552 


9.93 154 


9.71 622 


9.78 476 


0.21 524 


9.93 146 


22 


9.71 643 


9.78 505 


0.21 495 


9.93 138 


38 


23 


9.71 664 


9.78 533 


0.21 467 


9.93 131 


37 


24 


9.71 685 


9.78 562 


0.21 438 


9.93 123 


36 


25 


9.71 705 


9.78 590 


0.21 410 


9.93 115 


35 


26 


9.71 726 


9.78 618 


0.21 382 


9.93 108 


34 


27 


9.71 747 


9.78 647 


0.21 353 


9.93 100 


33 


28 


9.71 767 


9.78 675 


0.21 325 


9.93 092 


32 


29 
30 

31 


9.71 788 


9.78 704 


0.21 296 


9.93 084 


31 
30 

29 


9.71 809 


9.78 732 


0.21 268 


9.93 077 


9.71 829 


9.78 760 


0.21 240 


9.93 069 


32 


9.71 850 


9.78 789 


0.21 211 


9.93 061 


28 


33 


9.71 870 


9.78 817 


0.21 183 


9.93 053 


27 


34 


9.71 891 


9.78 845 


0.21 155 


9.93 046 


26 


35 


9.71 911 


9.78 874 


0.21 126 


9.93 038 


25 


36 


9.71 932 


9.78 902 


0.21 098 


9.93 030 


24 


37 


9.71 952 


9.78 930 


0.21 070 


9.93 022 


23 


38 


9.71 973 


9.78 959 


0.21 041 


9.93 014 


22 


39 
40 

41 


9.71 994 


9.78 987 


0.21 013 


9.93 007 


21 
20 

19 


9.72 014 


9.79 015 


0.20 985 


9.92 999 


9.72 034 


9.79 043 


0.20 957 


9.92 991 


42 


9.72 055 


9.79 072 


0.20 928 


9.92 983 


18 


43 


9.72 075 


9.79 100 


0.20 900 


9.92 976 


17 


44 


9.72 096 


9.79 '128 


0.20 872 


9.92 968 


16 


45 


9.72 116 


9.79 156 


0.20 844 


9.92 960 


15 


46 


9.72 137 


9.79 185 


0.20 815 


9.92 952 


14 


47 


9.72 157 


9.79 213 


0.20 787 


9.92 944 


13 


48 


9.72 177 


9.79 241 


0.20 759 


9.92 936 


12 


49 
50 

51 


9.72 198 


9.79 269 


0.20 731 


9.92 929 


11 
10 

9 


9.72 218 


9.79 297 


0.20 703 


9.92 921 


9.72 238 


9.79 326 


0.20 674 


9.92 913 


52 


9.72 259 


9.79 354 


0.20 646 


9.92 905 


8 


53 


9.72 279 


9.79 382 


0.20 618 


9.92 897 


7 


54 


9.72 299 


9.79 410 


0.20 590 


9.92 889 


6 


55 


9.72 320 


9.79 438 


0.20 562 


9.92 881 


5 


56 


9.72 340 


9.79 466 


0.20 534 


9.92 874 


4 


57 


9.72 360 


9.79 495 


0.20 505 


9.92 866 


3 


58 


9.72 381 


9.79 523 


0.20 477 


9.92 858 


2 


59 
60 


9.72 401 


9.79 551 


0.20 449 


9.92 850 


1 



9.72 421 


9.79 579 


0.20 421 


9.92 842 


. 


L. Cos. 


L. Cot. 


L. Tan. 


L. Sin. 


/ 



58 c 







—^ 



32 c 



59 



/ 


L. Sin. 


L. Tan. 


L. Cot. 


L. Cos. 






1 


9.72 421 


9.79 579 


0.20 421 


9.92 842 


60 

59 


9.72 441 


9.79 607 


0.20 393 


9.92 834 


2 


9.72 461 


9.79 635 


0.20 365 


9.92 826 


58 


3 


9.72 482 


9.79 663 


0.20 337 


9.92 818 


57 


4 


9.72 502 


9.79 691 


0.20 309 


9.92 810 


56 


5 


9.72 522 


9.79 719 


0.20 281 


9.92 803 


55 


6 


9.72 542 


9.79 747 


0.20 253 


9.92 795 


54 


7 


9.72 562 


9.79 776 


0.20 224 


9.92 787 


53 


8 


9.72 582 


9.79 804 


0.20 196 


9.92 779 


52 


9 
10 

11 


9.72 602 


9.79 832 


0.20 168 


9.92 771 


51 
50 

49 


9.72 622 


9.79 860 


0.20 140 


9.92 763 


9.72 643 


9.79 888 


0.20 112 


9.92 755 


12 


9.72 663 


9.79 916 


0.20 084 


9.92 747 


48 


13 


9.72 683 


9.79 944 


0.20 056 


9.92 739 


47 


14 


9.72 703 


9.79 972 


0.20 028 


9.92 731 


46 


15 


9.72 723 


9.80 000 


0.20 000 


9.92 723 


45 


16 


9.72 743 


9.80 028 


0.19 972 


9.92 715 


44 


17 


9.72 763 


9.80 056 


0.19 944 


9.92 707 


43 


18 


9.72 783 


9.80 084 


0.19 916 


9.92 699 


42 


19 
20 
21 


9.72 803 


9.80 112 


0.19 888 


9.92 691 


41 
40 

39 


9.72 823 


9.80 140 


0.19 860 


9.92 683 


9.72 843 


9.80 168 


0.19 832 


9.92 675 


22 


9.72 863 


9.80 195 


0.19 805 


9.92 667 


38 


23 


9.72 883 


9.80 223 


0.19 777 


9.92 659 


37 


24 


9.72 902 


9.80 251 


0.19 749 


9.92 651 


36 


25 


9.72 922 


9.80 279 


0.19 721 


9.92 643 


35 


26 


9.72 942 


9.80 307 


0.19 693 


9.92 635 


34 


27 


9.72 962 


9.80 335 


0.19 665 


9.92 627 


33 


28 


9.72 982 


9.80 363 


0.19 637 


9.92 619 


32 


29 
30 

31 


9.73 002 


9.80 391 


0.19 609 


9.92 611 


31 
30 

29 


9.73 022 


9.80 419 


0.19 581 


9.92 603 


9.73 041 


9.80 447 


0.19 553 


9.92 595 


32 


9.73 061 


9.80 474 


0.19 526 


9.92 587 


28 


33 


9.73 081 


9.80 502 


0.19 498 


9.92 579 


27 


34 


9.73 101 


9.80 530 


0.19 470 


9.92 571 


26 


35 


9.73 121 


9.80 558 


0.19 442 


9.92 563 


25 


36 


9.73 140 


9.80 586 


0.19 414 


9.92 555 


24 


37 


9.73 160 


9.80 614 


0.19 386 


9.92 546 


23 


38 


9.73 180 


9.80 642 


0.19 358 


9.92 538 


22 


39 
40 
41 


9.73 200 


9.80 669 


0.19 331 


9.92 530 


21 
20 

19 


9.73 219 


9.80 697 


0.19 303 


9.92 522 


9.73 239 


9.80 725 


0.19 275 


9.92 514 


42 


9.73 259 


9.80 753 


0.19 247 


9.92 506 


18 


43 


9.73 278 


9.80 781 


0.19 219 


9.92 498 


17 


44 


9.73 298 


9.80 808 


0.19 192 


9.92 490 


16 


45 


9.73 318 


9.80 836 


0.19 164 


9.92 482 


15 


46 


9.73 337 


9.80 864 


0.19 136 


9.92 473 


14 


47 


9.73 357 


9.80 892 


0.19 108 


9.92 465 


13 


48 


9.73 377 


9.80 919 


0.19 081 


9.92 457 


.12 


49 
50 

51 


9.73 396 


9.80 947 


0.19 053 


9.92 449 


11 
10 

9 


9.73 416 


9.80 975 


0.19 025 


9.92 441 


9.73 435 


9.81 003 


0.18 997 


9.92 433 


52 


9.73 455 


9.81 030 


0.18 970 


9.92 425 


8 


53 


9.73 474 


9.81 058 


0.18 942 


9.92 416 


7 


54 


9.73 494 


9.81 086 


0.18 914 


9.92 408 


6 


55 


9.73 513 


9.81 113 


0.18 887 


9.92 400 


5 


56 


9.73 533 


9.81 141 


0.18 859 


9.92 392 


4 


57 


9.73 552 


9.81 169 


0.18 831 


9.92 384 


3 


58 


9.73 572 


9.81 196 


0.18 804 


9.92 376 


2 


59 
60 


9.73 591 


9.81 224 


0.18 776 


9.92 367 


1 



9.73 611 


9.81 252 


0.18 748 


9.92 359 




L. Cos. 


L. Cot. 


L. Tan. 


L. Sin. 


t 



57' 



6o 



33 



, 


L. Sin. 


L. Tan. 


L. Cot. 


L. Cos. 






1 


9.73 611 


9.81 252 


0.18 748 


9.92 359 


60 

59 


9.73 630 


9.81 279 


0.18 721 


9.92 351 


2 


9.73 650 


9.81 307 


0.18 693 


9.92 343 


58 


3 


9.73 669 


9.81 335 


0.18 665 


9.92 335 


57 


4 


9.73 689 


9.81 362 


0.18 638 


9.92 326 


56 


5 


9.73 708 


9.81 390 


0.18 610 


9.92 318 


55 


6 


9.73 727 


9.81 418 


0.18 582 


9.92 310 


54 


7 


9.73 747 


9.81 445 


0.18 555 


9.92 302 


53 


8 


9.73 766 


9.81 473 


0.18 527 


9.92 293 


52 


9 
10 
11 


9.73 785 


9.81 500 


0.18 500 


9.92 285 


51 
50 

49 


9.73 805 


9.81 528 


0.18 472 


9.92 277 


9.73 824 


9.81 556 


0.18 444 


9.92 269 


12 


9.73 843 


9.81 583 


0.18 417 


9.92 260 


48 


13 


9.73 863 


9.81 611 


0.18 389 


9.92 252 


47 


14 


9.73 882 


9.81 638 


0.18 362 


9.92 244 


46 


15 


9.73 901 


9.81 666 


0.18 334 


9.92 235 


45 


16 


9.73 921 


9.81 693 


0.18 307 


9.92 227 


44 


17 


9.73 940 


9.81 721 


0.18 279 


9.92 219 


43 


18 


9.73 959 


9.81 748 


0.18 252 


9.92 211 


42 


19 
20 
21 


9.73 978 


9.81 776 


0.18 224 


9.92 202 


41 
40 

39 


9.73 997 


9.81 803 


0.18 197 


9.92 194 


9.74 017 


9.81 831 


0.18 169 


9.92 186 


22 


9.74 036 


9.81 858 


0.18 142 


9.92 177 


38 


23 


9.74 055 


9.81 886 


0.18 114 


9.92 169 


37 


24 


9.74 074 


9.81 913 


0.18 087 


9.92 161 


36 


25 


9.74 093 


9.81 941 


0.18 059 


9.92 152 


35 


26 


9.74 113 


9.81 968 


0.18 032 


9.92 144 


34 


27 


9.74 132 


9.81 996 


0.18 004 


9.92 136 


33 


28 


9.74 151 


9.82 023 


0.17 977 


9.92 127 


32 


29 
30 

31 


9.74 170 


9.82 051 


0.17 949 


9.92 119 


31 
30 
29 


9.74 189 


9.82 078 


0.17 922 


9.92 111 


9.74 208 


9.82 106 


0.17 894 


9.92 102 


32 


9.74 227 


9.82 133 


0.17 867 


9.92 094 


28 


33 


9.74 246 


9.82 161 


0.17 839 


9.92 086 


27 


34 


9.74 265 


9.82 188 


0.17 812 


9.92 077 


26 


35 


9.74 284 


9.82 215 


0.17 785 


9.92 069 


25 


36 


9.74 303 


9.82 243 


0.17 757 


9.92 060 


24 


37 


9.74 322 


9.82 270 


0.17 730 


9.92 052 


23 


38 


9.74 341 


9.82 298 


0.17 702 


9.92 044 


22 


39 
40 
41 


9.74 360 


9.82 325 


0.17 675 


9.92 035 


21 
20 

19 


9.74 379 


9.82 352 


0.17 648 


9.92 027 


9.74 398 


9.82 380 


0.17 620 


9.92 018 


42 


9.74 417 


9.82 407 


0.17 593 


9.92 010 


18 


43 


9.74 436 


9.82 435 


0.17 565 


9.92 002 


17 


44 


9.74 455 


9.82 462 


0.17 538 


9.91 993 


16 


45 


9.74 474 


9.82 489 


0.17 511 


9.91 985 


15 


46 


9.74 493 


9.82 517 


0.17 483 


9.91 976 


14 


47 


9.74 512 


9.82 544 


0.17 456 


9.91 968 


13 


48 


9.74 531 


9.82 571 


0.17 429 


9.91 959 


12 


49 
50 
51 


9.74 549 


9.82 599 


0.17 401 


9.91 951 


11 
10 

9 


9.74 568 


9.82 626 


0.17 374 


9.91 942 


9.74 587 


9.82 653 


0.17 347 


9.91 934 


52 


9.74 606 


9.82 681 


0.17 319 


9.91 925 


8 


53 


9.74 625 


9.82 708 


0.17 292 


9.91 917 


7 


54 


9.74 644 


9.82 735 


0.17 265 


9.91 908 


6 


55 


9.74 662 


9.82 762 


0.17 238 


9.91 900 


5 


56 


9.74 681 


9.82 790 


0.17 210 


9.91 891 


4 


57 


9.74 700 


9.82 817 


0.18 183 


9.91 883 


3 


58 


9.74 719 


9.82 844 


0.17 156 


9.91 874 


2 


59 
60 


9.74 737 


9.82 871 


0.17 129 


9.91 866 


1 



9.74 756 


9.82 899 


0.17 101 


9.91 857 




L. Cos. 


L. Cot. 


L. Tan. 


L. Sin. 


/ 



56 



^mmm 



34' 



6i 



t 


L. Sin. 


L. Tan. 


L. Cot. 


L. Cos. 






1 


9.74 756 


9.82 899 


0.17 101 


9.91 857 


60 

59 


9.74 775 


9.82 926 


0.17 074 


9.91 849 


2 


9.74 794 


9.82 953 


0.17 047 


9.91 840 


58 


3 


9.74 812 


9.82 980 


0.17 020 


9.91 832 


57 


4 


9.74 831 


9.83 008 


0.16 992 


9.91 823 


56 


5 


9.74 850 


9.83 035 


0.16 965 


9.91 815 


55 


6 


9.74 868 


9.83 062 


0.16 938 


9.91 806 


54 


7 


9.74 887 


9.83 089 


0.16 911 


9.91 798 


53 


8 


9.74 906 


9.83 117 


0.16 883 


9.91 789 


52 


9 
10 

11 


9.74 924 


9.83 144 


0.16 856 


9.91 781 


51 
50 

49 


9.74 943 


9.83 171 


0.16 829 


9.91 772 


9.74 961 


9.83 198 


0.16 802 


9.91 763 


12 


9.74 980 


9.83 225 


0.16 775 


9.91 755 


48 


13 


9.74 999 


9.83 252 


0.16 748 


9.91 746 


47 


14 


9.75 017 


9.83 280 


0.16 720 


9.91 738 


46 


15 


9.75 036 


9.83 307 


0.16 693 


9.91 729 


45 


16 


9.75 054 


9.83 334 


0.16 666 


9.91 720 


44 


17 


9.75 073 


9.83 361 


0.16 639 


9.91 712 


43 


18 


9.75 091 


9.83 388 


0.16 612 


9.91 703 


42 


19 
20 

21 


9.75 110 


9.83 415 


0.16 585 


9.91 695 


41 
40 

39 


9.75 128 


9.83 442 


0.16 558 


9.91 686 


9.75 147 


9.83 470 


0.16 530 


9.91 677 


22 


9.75 165 


9.83 497 


0.16 503 


9.91 669 


38 


23 


9.75 184 


9.83 524 


0.16 476 


9.91 660 


37 


24 


9.75 202 


9.83 551 


0.16 449 


9.91 651 


36 


25 


9.75 221 


9.83 578 


0.16 422 


9.91 643 


35 


26 


9.75 239 


9.83 605 


0.16 395 


9.91 634 


34 


27 


9.75 258 


9.83 632 


0.16 368 


9.91 625 


33 


28 


9.75 276 


9.83 659 


0.16 341 


9.91 617 


32 


29 
30 
31 


9.75 294 


9.83 686 


0.16 314 


9.91 608 


31 
30 

29 


9.75 313 


9.83 713 


0.16 287 


9.91 599 


9.75 331 


9.83 740 


0.16 260 


9.91 591 


32 


9.75 350 


9.83 768 


0.16 232 


9.91 582 


28 


33 


9.75 368 


9.83 795 


0.16 205 


9.91 573 


27 


34 


9.75 386 


9.83 822 


0.16 178 


9.91 565 


26 


35 


9.75 405 


9.83 849 


0.16 151 


9.91 556 


25 


36 


9.75 423 


9.83 876 


0.16 124 


9.91 547 


24 


37 


9.75 441 


9.83 903 


0.16 097 


9.91 538 


23 


38 


9.75 459 


9.83 930 


0.16 070 


9.91 530 


22 


39 
40 
41 


9.75 478 


9.83 957 


0.16 043 


9.91 521 


21 
20 

19 


9.75 496 


9.83 984 


0.16 016 


9.91 512 


9.75 514 


9.84 Oil 


0.15 989 


9.91 504 


42 


9.75 533 


9.84 038 


0.15 962 


9.91 495 


18 


43 


9.75 551 


9.84 065 


0.15 935 


9.91 486 


17 


44 


9.75 569 


9.84 092 


0.15 908 


9.91 477 


16 


45 


9.75 587 


9.84 119 


0.15 881 


9.91 469 


15 


46 


9.75 605 


9.84 146 


0.15 854 


9.91 460 


14 


47 


9.75 624 


9.84 173 


0.15 827 


9.91 451 


13 


48 


9.75 642 


9.84 200 


0.15 800 


9.91442 


12 


49 
50 

51 


9.75 660 


9.84 227 


0.15 773 


9.91 433 


11 
10 

9 


9.75 678 


9.84 254 


0.15 746 


9.91 425 


9.75 696 


9.84 280 


0.15 720 


9.91 416 


52 


9.75 714 


9.84 307 


0.15 693 


9.91 407 


8 


53 


9.75 733 


9.84 334 


0.15 666 


9.91 398 


7 


54 


9.75 751 


9.84 361 


0.15 639 


9.91 389 


6 


55 


9.75 769 


9.84 388 


0.15 612 


9.91 381 


5 


56 


9.75 787 


9.84 415 


0.15 585 


9.91 372 


4 


57 


9.75 805 


9.84 442 


0.15 558 


9.91 363 


3 


58 


9.75 823 


9.84 469 


0.15 531 


9.91 354 


2 


59 
60 


9.75 841 


9.84 496 


0.15 504 


9.91 345 


1 



9.75 859 


9.84 523 


0.15 477 


9.91 336 




L. Cos. 


L. Cot. 


L. Tan. 


L. Sin. 


t 



55 



62 



35' 



t 


L. Sin. 


L. Tan. 


L. Cot. 


L. Cos. 






1 


9.75 859 


9.84 523 


0.15 477 


9.91 336 


60 

59 


9.75 877 


9.84 550 


0.15 450 


9.91 328 


2 


9.75 895 


9.84 576 


0.15 424 


9.91 319 


58 


3 


9.75 913 


9.84 603 


0.15 397 


9.91 310 


57 


4 


9.75 931 


9.84 630 


0.15 370 


9.91 301 


56 


5 


9.75 949 


9.84 657 


0.15 343 


9.91 292 


55 


6 


9.75 967 


9.84 684 


0.15 316 


9.91 283 


54 


7 


9.75 985 


9.84 711 


0.15 289 


9.91 274 


53 


8 


9.76 003 


9.84 738 


0.15 262 


9.91 266 


52 


9 
10 
11 


9.76 021 


9.84 764 


0.15 236 


9.91 257 


51 
50 

49 


9.76 039 


9.84 791 


0.15 209 


9.91 248 


9.76 057 


9.84 818 


0.15 182 


9.91 239 


12 


9.76 075 


9.84 845 


0.15 155 


9.91 230 


48 


13 


9.76 093 


9.84 872 


0.15 128 


9.91 221 


47 


14 


9.76 111 


9.84 899 


0.15 101 


9.91 212 


46 


15 


9.76 129 


9.84 925 


0.15 075 


9.91 203 


45 


16 


9.76 146 


9.84 952 


0.15 048 


9.91 194 


44 


17 


9.76 164 


9.84 979 


0.15 021 


9.91 185 


43 


18 


9.76 182 


9.85 006 


0.14 994 


9.91 176 


42 


19 
20 

21 


9.76 200 


9.85 033 


0.14 967 


9.91 167 


41 
40 

39 


9.76 218 


9.85 059 


0.14 941 


9.91 158 


9.76 236 


9.85 086 


0.14 914 


9.91 149 


22 


9.76 253 


9.85 113 


0.14 887 


9.91 141 


38 


23 


9.76 271 


9.85 140 


0.14 860 


9.91 132 


37 


24 


9.76 289 


9.85 166 


0.14 834 


9.91 123 


36 


25 


9.76 307 


9.85 193 


0.14 807 


9.91 114 


35 


26 


9.76 324 


9.85 220 


0.14 780 


9.91 105 


34 


27 


9.76 342 


9.85 247 


0.14 753 


9.91 096 


33 


28 


9.76 360 


9.85 273 


0.14 72? 


9.91 087 


32 


29 
30 

31 


9.76 378 


9.85 300 


0.14 700 


9.91 078 


31 
30 

29 


9.76 395 


9.85 327 


0.14 673 


9.91 069 


9.76 413 


9.85 354 


0.14 646 


9.91 060 


32 


9.76 431 


9.85 330 


0.14 620 


9.91 051 


28 


33 


9.76 448 


9.85 407 


0.14 593 


9.91 042 


27 


34 


9.76 466 


9.85 434 


0.14 566 


9.91 033 


26 


35 


9.76 484 


9.85 460 


0.14 540 


9.91 023 


25 


36 


9.76 501 


9.85 487 


0.14 513 


9.91 014 


24 


37 


9.76 519 


9.85 514 


0.14 486 


9.91 005 


23 


38 


9.76 537 


9.85 540 


0.14 460 


9.90 996 


22 


39 
40 
41 


9.76 554 


9.85 567 


0.14 433 


9.90 987 


21 
20 

19 


9.76 572 


9.85 594 


0.14 406 


9.90 978 


9.76 590 


9.85 620 


0.14 380 


9.90 969 


42 


9.76 607 


9.85 647 


0.14 353 


9.90 960 


18 


43 


9.76 625 


9.85 674 


0.14 326 


9.90 951 


17 


44 


9.76 642 


9.85 700 


0.14 300 


9.90 942 


16 


45 


9.76 660 


9.85 727 


0.14 273 


9.90 933 


15 


46 


9.76 677 


9.85 754 


0.14 246 


9.90 924 


14 


47 


9.76 695 


9.85 780 


0.14 220 


9.90 915 


13 


48 


9.76 712 


9.85 807 


0.14 193 


9.90 906 


12 


49 
50 
51 


9.76 730 


9.85 834 


0.14 166 


9.90 896 


11 
10 

9 


9.76 747 


9.85 860 


0.14 140 


9.90 887 


9.76 765 


9.85 887 


0.14 113 


9.90 878 


52 


9.76 782 


9.85 913 


0.14 087 


9.90 869 


8 


53 


9.76 800 


9.85 940 


0.14 060 


9.90 860 


7 


54 


9.76 817 


9.85 967 


0.14 033 


9.90 851 


6 


55 


9.76 835 


9.85 993 


0.14 007 


9.90 842 


5 


56 


9.76 852 


9.86 020 


0.13 980 


9.90 832 


4 


57 


9.76 870 


9.86 046 


0.13 954 


9.90 823 


3 


58 


9.76 887 


9.86 073 


0.13 927 


9.90 814 


2 


59 
60 


9.76 904 


9.86 100 


0.13 900 


9.90 805 


1 



9.76 922 


9.86 126 


0.13 874 


9.90 796 




L. Cos. 


L. Cot. 


L. Tan. 


L. S'n. 


/ 



54 c 



^m 



36 



63 



/ 


L. Sin. 


L. Tan. 


L. Cot. 


L. Cos. 






1 


9.76 922 


9.86 126 


0.13 874 


9.90 796 


60 

59 


9.76 939 


9.86 153 


0.13 847 


9.90 787 


2 


9.76 957 


9.86 179 


0.13 821 


9.90 777 


58 


3 


9.76 974 


9.86 206 


0.13 794 


9.90 768 


57 


4 


9.76 991 


9.86 232 


0.13 768 


9.90 759 


56 


5 


9.77 009 


9.86 259 


0.13 741 


9.90 750 


55 


6 


9.77 026 


9.86 285 


0.13 715 


9.90 741 


54 


7 


9.77 043 


9.86 312 


0.13 688 


9.90 731 


53 


8 


9.77 061 


9.86 338 


0.13 662 


9.90 722 


52 


9 
10 

11 


9.77 078 


9.86 365 


0.13 635 


9.90 713 


51 
50 

49 


9.77 095 


9.86 392 


0.13 608 


9.90 704 


9.77 112 


9.86 418 


0.13 582 


9.90 694 


12 


9.77 130 


9.86 445 


0.13 555 


9.90 685 


48 


13 


9.77 147 


9.86 471 


0.13 529 


9.90 676 


47 


14 


9.77 164 


9.86 498 


0.13 502 


9.90 667 


46 


15 


9.77 181 


9.86 524 


0.13 476 


9.90 657 


45 


16 


9.77 199 


9.86 551 


0.13 449 


9.90 648 


44 


17 


9.77 216 


9.86 577 


0.13 423 


9.90 639 


43 


18 


9.77 233 


9.86 603 


0.13 397 


9.90 630 


42 


19 
20 
21 


9.77 250 


9.86 630 


0.13 370 


9.90 620 


41 
40 

39 


9.77 268 


9.86 656 


0.13 344 


9.90 611 


9.77 285 


9.86 683 


0.13 317 


9.90 602 


22 


9.77 302 


9.86 709 


0.13 291 


9.90 592 


38 


23 


9.77 319 


9.86 736 


0.13 264 


9.90 583 


37 


24 


9.77 336 


9.86 762 


0.13 238 


9.90 574 


36 


25 


9.77 353 


9.86 789 


0.13 211 


9.90 565 


35 


26 


9.77 370 


9.86 815 


0.13 185 


9.90 555 


34 


27 


9.77 387 


9.86 842 


0.13 158 


9.90 546 


33 


28 


9.77 405 


9.86 868 


0.13 132 


9.90 537 


32 


29 
30 

31 


9.77 422 


9.86 894 


0.13 106 


9.90 527 


31 
30 

29 


9.77 439 


9.86 921 


0.13 079 


9.90 518 


9.77 456 


9.86 947 


0.13 053 


9.90 509 


32 


9.77 473 


9.86 974 


0.13 026 


9.90 499 


28 


33 


9.77 490 


9.87 000 


0.13 000 


9.90 490 


27 


34 


9.77 507 


9.87 027 


0.12 973 


9.90 480 


26 


35 


9.77 524 


9.87 053 


0.12 947 


9.90 471 


25 


36 


9.77 541 


9.87 079 


0.12 921 


9.90 462 


24 


37 


9.77 558 


9.87 106 


0.12 894 


9.90 452 


23 


38 


9.77 575 


9.87 132 


0.12 868 


9.90 443 


22 


39 
40 
41 


9.77 592 


9.87 158 


0.12 842 


9.90 434 


21 
20 

19 


9.77 609 


9.87 185 


0.12 815 


9.90 424 


9.77 626 


9.87 211 


0.12 789 


9.90 415 


42 


9.77 643 


9.87 238 


0.12 762 


9.90 405 


18 


43 


9.77 660 


9.87 264 


0.12 736 


9.90 396 


17 


44 


9.77 677 


9.87 290 


0.12 710 


9.90 386 


16 


45 


9.77 694 


9.87 317 


0.12 683 


9.90 377 


15 


46 


9.77 711 


9.87 343 


0.12 657 


9.90 368 


14 


47 


9.77 728 


9.87 369 


0.12 631 


9.90 358 


13 


48 


9.77 744 


9.87 396 


0.12 604 


9.90 349 


12 


49 
50 
51 


9.77 761 


9.87 422 


0.12 578 


9.90 339 


11 
10 

9 


9.77 778 


9.87 448 


0.12 552 


9.90 330 


9.77 795 


9.87 475 


0.12 525 


9.90 320 


52 


9.77 812 


9.87 501 


0.12 499 


9.90 311 


8 


53 


9.77 829 


9.87 527 


0.12 473 


9.90 301 


7 


54 


9.77 846 


9.87 554 


0.12 446 


9.90 292 


6 


55 


9.77 862 


9.87 580 


0.12 420 


9.90 282 


5 


56 


9.77 879 


9.87 606 


0.12 394 


9.90 273 


4 


57 


9.77 896 


9.87 633 


0.12 367 


9.90 263 


3 


58 


9.77 913 


9.87 659 


0.12 341 


9.90 254 


2 


59 
60 


9.77 930 


9.87 685 


0.12 315 


9.90 244 


1 



9.77 946 


9.87 711 


0.12 289 


9.90 235 




L. Cos. 


L. Cot. 


L. Tan. 


L. Sin. 


t 



53 



6 4 



37° 



t 


L. Sin. 


L. Tan. 


L. Cot. 


L. Cos. 


1 




1 


9.77 946 


9.87 711 


0.12 289 


9.90 235 


60 1 


9.77 963 


9.87 738 


0.12 262 


9.90 225 


59 


2 


9.77 980 


9.87 764 


0.12 236 


G.90 216 


58 


3 


9.77 997 


9.87 790 


0.12 210 


9.90 206 


57 


4 


9.78 013 


9.87 817 


0.12 183 


9.90 197 


56 


5 


9.78 030 


9.87 843 


0.12 157 


9.90 187 


55 


6 


9.78 047 


9.87 869 


0.12 131 


9.90 178 


54 


7 


9.78 063 


9.87 895 


0.12 105 


9.90 168 


53 


8 


9.78 080 


9.87 922 


0.12 078 


9.90 159 


52 


9 
10 

11 


9.78 097 


9.87 948 


0.12 052 


9.90 149 


51 
50 

49 


9.78 113 


9.87 974 


0.12 026 


9.90 139 


9.78 130 


9.88 000 


0.12 000 


9.90 130 


12 


9.78 147 


9.88 027 


0.11 973 


9.90 120 


48 


13 


9.78 163 


9.88 053 


0.11 947 


9.90 111 


47 


14 


9.78 180 


9.88 079 


0.11 921 


9.90 101 


46 


15 


9.78 197 


9.88 105 


0.11 895 


9.90 091 


45 


16 


9.78 213 


9.88 131 


0.11 869 


9.90 082 


44 


17 


9.78 230 


9.88 158 


0.11 842 


9.90 072 


43 


18 


9.78 246 


9.88 184 


0.11 816 


9.90 063 


42 


19 
20 

21 


9.78 263 


9.88 210 


0.11 790 


9.90 053 


41 
40 
39 


9.78 280 


9.88 236 


0.11 764 


9.90 043 


9.78 296 


9.88 262 


0.11 738 


9.90 034 


22 


9.78 313 


9.88 289 


0.11 711 


9.90 024 


38 


23 


9.78 329 


9.88 315 


0.11 685 


9.90 014 


37 


24 


9.78 346 


9.88 341 


0.11 659 


9.90 005 


36 


25 


9.78 362 


9.88 367 


0.11 633 


9.89 995 


35 


26 


9.78 379 


9.88 393 


0.11 607 


9.89 985 


34 


27 


9.78 395 


9.88 420 


0.11 580 


9.89 976 


33 


28 


9.78 412 


9.88 446 


0.11 554 


9.89 966 


32 


29 
30 

31 


9.78 428 


9.88 472 


0.11 528 


9.89 956 


31 
30 

29 


9.78 445 


9.88 498 


0.11 502 


9.89 947 


9.78 461 


9.88 524 


0.11 476 


9.89 937 


32 


9.78 478 


9.88 550 


0.11 450 


9.89 927 


28 


33 


9.78 494 


9.88 577 


0.11 423 


9.89 918 


27 


34 


9.78 510 


9.88 603 


0.11 397 


9.89 908 


26 


35 


9.78 527 


9.88 629 


0.11 371 


9.89 898 


25 


36 


9.78 543 


9.88 655 


0.11 345 


9.89 888 


24 


37 


9.78 560 


9.88 681 


0.11 319 


9.89 879 


23 


38 


9.78 576 


9.88 707 


0.11 293 


9.89 869 


22 


39 
40 
41 


9.78 592 


9.88 733 


0.11 267 


9.89 859 


21 
20 
19 


9.78 609 


9.88 759 


0.11 241 


9.89 849 


9.78 625 


9.88 786 


0.11 214 


9.89 840 


42 


9.78 642 


9.88 812 


0.11 188 


9.89 830 


18 


43 


9.78 658 


9.88 838 


0.11 162 


9.89 820 


17 


44 


9.78 674 


9.88 864 


0.11 136 


9.89 810 


16 


45 


9.78 691 


9.88 890 


0.11 110 


9.89 801 


15 


46 


9.78 707 


9.88 916 


0.11 084 


9.89 791 


14 


47 


9.78 723 


9.88 942 


0.11 058 


9.89 781 


13 


48 


9.78 739 


9.88 968 


0.11 032 


9.89 771 


12 


49 
50 

51 


9.78 756 


9.88 994 


0.11 006 


9.89 761 


11 
10 

9 


9.78 772 


9.89 020 


0.10 980 


9.89 752 


9.78 788 


9.89 046 


0.10 954 


9.89 742 


52 


9.78 805 


9.89 073 


0.10 927 


9.89 732 


8 


53 


9.78 821 


9.89 099 


0.10 901 


9.89 722 


7 


54 


9.78 837 


9.89 125 


0.10 875 


9.89 712 


6 


55 


9.78 853 


9.89 151 


0.10 849 


9.89 702 


5 


56 


9.78 869 


9.89 177 


0.10 823 


9.89 693 


4 


57 


9.78 886 


9.89 203 


0.10 797 


9.89 683 


3 


58 


9.78 902 


9.89 229 


0.10 771 


9.89 673 


2 


59 
60 


9.78 918 


9.89 255 


0.10 745 


9.89 663 


1 



9.78 934 


9.89 281 


0.10 719 


9.89 653 




L. Cos. 


L. Cot. 


L. Tan. 


L. Sin. 


' 



52 c 











38' 



65 



t 


L. Sin. 


L. Tan. 


L. Cot. 


L. Cos. 






1 


9.78 934 


9.89 281 


0.10 719 


9.89 653 


60 

59 


9.78 950 


9.89 307 


0.10 693 


9.89 643 


2 


9.78 967 


9.89 333 


0.10 667 


9.89 633 


58 


3 


9.78 983 


9.89 359 


0.10 641 


9.89 624 


57 


4 


9.78 999 


9.89 385 


0.10 615 


9.89 614 


56 


5 


9.79 015 


9.89 411 


0.10 589 


9.89 604 


55 


6 


9.79 031 


9.89 437 


0.10 563 


9.89 594 


54 


7 


9.79 047 


9.89 463 


0.10 537 


9.89 584 


53 


8 


9.79 063 


9.89 489 


0.10 511 


9.89 574 


52 


9 
10 

11 


9.79 079 


9.89 515 


0.10 485 


9.89 564 


51 
50 

49 


9.79 095 


9.89 511 


0.10 459 


9.89 554 


9.79 111 


9.89 567 


0.10 433 


9.89 544 


12 


9.79 128 


9.89 593 


0.10 407 


9.89 534 


48 


13 


9.79 144< 


9.89 619 


0.10 381 


9.89 524 


47 


14 


9.79 160 


9.89 645 


0.10 355 


9.89 514 


46 


15 


9.79 176 


9.89 671 


0.10 329 


9.89 504 


45 


16 


9.79 192 


9.89 697 


0.10 303 


9.89 495 


44 


17 


9.79 208 


9.89 723 


0.10 277 


9.89 485 


43 


18 


9.79 224 


9.89 749 


0.10 251 


9.89 475 


42 


19 
20 
21 


9.79 240 


9.89 775 


0.10 225 


9.89 465 


41 
40 

39 


9.79 256 


9.89 801 


0.10 199 


9.89 455 


9.79 272 


9.89 827 


0.10 173 


9.89 445 


22 


9.79 288 


9.89 853 


0.10 147 


9.89 435 


38 


23 


9.79 304 


9.89 879 


0.10 121 


9.89 425 


37 


24 


9.79 319 


9.89 905 


0.10 095 


9.89 415 


36 


25 


9.79 335 


9.89 931 


0.10 069 


9.89 405 


35 


26 


9.79 351 


9.89 957 


0.10 043 


9.89 395 


34 


27 


9.79 367 


9.89 983 


0.10 017 


9.89 385 


33 


28 


9.79 383 


9.90 009 


0.09 991 


9.89 375 


32 


29 
30 

31 


9.79 399 


9.90 035 


0.09 965 


9.89 364 


31 
30 

29 


9.79 415 


9.90 061 


0.09 939 


9.89 354 


9.79 431 


9.90 086 


0.09 914 


9.89 344 


32 


9.79 447 


9.90 112 


0.09 888 


9.89 334 


28 


33 


9.79 463 


9.90 138 


0.09 862 


9.89 324 


27 


34 


9.79 418 


9.90 164 


0.09 836 


9.89 314 


26 


35 


9.79 494 


9.90 190 


0.09 810 


9.89 304 


25 


36 


9.79 510 


9.90 216 


0.09 784 


9.89 294 


24 


37 


9.79 526 


9.90 242 


0.09 758 


9.89 284 


23 


38 


9.79 542 


9.90 268 


0.09 732 


9.89 274 


22 


39 
40 

41 


9.79 558 


9.90 294 


0.09 706 


9.89 264 


21 
20 

19 


9.79 573 


9.90 320 


0.09 680 


9.89 254 


9.79 589 


9.90 316 


0.09 654 


9.89 244 


42 


9.79 605 


9.90 371 


0.09 629 


9.89 233 


18 


43 


9.79 621 


9.90 397 


0.09 603 


9.89 223 


17 


44 


9.79 636 


9.90 423 


0.09 577 


9.89 213 


16 


45 


9.79 652 


9.90 449 


0.09 551 


9.89 203 


15 


46 


9.79 668 


9.90 475 


0.09 525 


9.89 193 


14 


47 


9.79 684 


9.90 501 


0.09 499 


9.89 183 


13 


48 


9.79 699 


9.90 527 


0.09 473 


9.89 173 


12 


49 
50 
51 


9.79 715 


9.90 553 


0.09 447 


9.89 162 


11 
10 

9 


9.79 731 


9.90 578 


0.09 422 


9.89 152 


9.79 746 


9.90 604 


0.09 396 


9.89 142 


52 


9.79 762 


9.90 630 


0.09 370 


9.89 132 


8 


53 


9.79 778 


9.90 656 


0.09 344 


9.89 122 


7 


54 


9.79 793 


9.90 682 


0.09 318 


9.89 112 


6 


55 


9.79 809 


9.90 708 


0.09 292 


9.89 101 


5 


56 


9.79 825 


9.90 731 


0.09 266 


9.89 091 


4 


57 


9.79 840 


9.90 759 


0.09 241 


9.89 081 


3 


58 


9.79 856 


9.90 785 


0.09 215 


9.89 071 


2 


59 
60 


9.79 872 


9.90 811 


0.09 189 


9.89 060 


1 



9.79 887 


9.90 837 


0.09 163 


9.89 050 




L. Cos. 


L. Cot. 


L. Tan. 


L. Sin. 


/ 



51 



66 



39 



i 


L. Sin. 


L. Tan. 


L. Cot. 


L. Cos. 






1 


9.79 887 


9.90 837 


0.09 163 


9.89 050 


60 

59 


9.79 903 


9.90 863 


0.09 137 


9.89 040 


2 


9.79 918 


9.90 889 


0.09 111 


9.89 030 


58 


3 


9.79 934 


9.90 914 


0.09 086 


9.89 020 


57 


4 


9.79 950 


9.90 940 


0.09 060 


9.89 009 


56 


5 


9.79 965 


9.90 966 


0.09 034 


9.88 999 


55 


6 


9.79 981 


9.90 992 


0.09 008 


9.88 989 


54 


7 


9.79 996 


9.91 018 


0.08 982 


9.88 978 


53 


8 


9.80 012 


9.91 043 


0.08 957 


9.88 968 


52 


9 
10 

11 


9.80 027 


9.91 069 


0.08 931 


9.88 958 


51 
50 

49 


9.80 043 


9.91 095 


0.08 905 


9.88 948 


9.80 058 


9.91 121 


0.08 879 


9.88 937 


12 


9.80 074 


9.91 147 


0.08 853 


9.88 927 


48 


13 


9.80 089 


9.91 172 


0.08 828 


9.88 917 


47 


14 


9.80 105 


9.91 198 


0.08 802 


9.88 906 


46 


15 


9.80 120 


9.91 224 


0.08 776 


9.88 896 


45 


16 


9.80 136 


9.91 250 


0.08 750 


9.88 886 


44 


17 


9.80 151 


9.91 276 


0.08 724 


9.88 875 


43 


18 


9.80 166 


9.91 301 


0.08 699 


9.88 865 


42 


19 
20 
21 


9.80 182 


9.91 327 


0.08 673 


9.88 855 


41 
40 

39 


9.80 197 


9.91 353 


0.08 647 


9.88 844 


9.80 213 


9.91 379 


0.08 621 


9.88 834 


22 


9.80 228 


9.91 404 


0.08 596 


9.88 824 


38 


23 


9.80 244 


9.91 430 


0.08 570 


9.88 813 


37 


24 


9.80 259 


9.91 456 


0.08 544 


9.88 803 


36 


25 


9.80 274 


9.91 482 


0.08 518 


9.88 793 


35 


26 


9.80 290 


9.91 507 


0.08 493 


9.88 782 


34 


27 


9.80 305 


9.91 533 


0.08 467 


9.88 772 


33 


28 


9.80 320 


9.91 559 


0.08 441 


9.88 761 


32 


29 
30 

31 


9.80 336 


9.91 585 


0.08 415 


9.88 751 


31 
30 

29 


9.80 351 


9.91 610 


0.08 390 


9.88 741 


9.80 366 


9.91 636 


0.08 364 


9.88 730 


32 


9.80 382 


9.91 662 


0.08 338 


9.88 720 


28 


33 


9.80 397 


9.91 688 


0.08 312 


9.88 709 


27 


34 


9.80 412 


9.91 713 


0.08 287 


9.88 699 


26 


35 


9.80 428 


9.91 739 


0.08 261 


9.88 688 


25 


36 


9.80 443 


9.91 765 


0.08 235 


9.88 678 


24 


37 


9.80 458 


9.91 791 


0.08 209 


9.88 668 


23 


38 


9.80 473 


9.91 816 


0.08 184 


9.88 657 


22 


39 
40 

41 


9.80 489 


9.91 842 


0.08 158 


9.88 647 


21 
20 

19 


9.80 504 


9.91 868 


0.08 132 


9.88 636 


9.80 519 


9.91 893 


0.08 107 ' 


9.88 626 


42 


9.80 534 


9.91 919 


0.08 081 


9.88 615 


18 


43 


9.80 550 


9.91 945 


0.08 055 


9.88 605 


17 


44 


9.80 565 


9.91 971 


0.08 029 


9.88 594 


16 


45 


9.80 580 


9.91 996 


0.08 004 


9.88 584 


15 


46 


9.80 595 


9.92 022 


0.07 978 


9.88 573 


14 


47 


9.80 610 


9.92 048 


0.07 952 


9.88 563 


13 


48 


9.80 625 


9.92 073 


0.07 927 


9.88 552 


12 


49 
50 

51 


9.80 641 


9.92 099 


0.07 901 


9.88 542 


11 
10 

9 


9.80 656 


9.92 125 


0.07 875 


9.88 531 


9.80 671 


9.92 150 


0.07 850 


9.88 521 


52 


9.80 686 


9.92 176 


0.07 824 


9.88 510 


8 


53 


9.80 701 


9.92 202 


0.07 798 • 


9.88 499 


7 


54 


9.80 716 


9.92 227 


0.07 773 


9.88 489 


6 


55 


9.80 731 


9.92 253 


0.07 747 


9.88 478 


5 


56 


9.80 746 


9.92 279 


0.07 721 


9.88 468 


4 


57 


9.80 762 


9.92 304 


0.07 696 


9.88 457 


3 


58 


9.80 777 


9.92 330 


0.07 670 


9.88 447 


2 


59 
60 


9.80 792 


9.92 356 


0.07 644 


9.88 436 


1 



9.80 807 


9.92 381 


0.07 619 


9.88 425 




L. Cos. 


L, Cot. 


L. Tan. 


L. Sin. 


9 



50 



4(y 



6 7 



t 


L. Sin. 


L. Tan. 


L. Cot. 


L. Cos. 


60 

59 




1 


9.80 807 


9.92 381 


0.07 619 


9.88 425 


9.80 822 


9.92 407 


0.07 593 


9.88 415 


2 


9.80 837 


9.92 433 


0.07 567 


9.88 404 


58 


3 


9.80 852 


9.92 458 


0.07 542 


9.88 394 


57 


4 


9.80 867 


9.92 484 


0.07 516 


9.88 383 


56 


5 


9.80 882 


9.92 510 


0.07 490 


9.88 372 


55 


6 


9.80 897 


9.92 535 


0.07 465 


9.88 362 


54 


7 


9.80 912 


9.92 561 


0.07 439 


9.88 351 


53 


8 


9.80 927 


9.92 587 


0.07 413 


9.88 340 


52 


9 
10 

11 


9.80 942 


9.92 612 


0.07 388 


9.88 330 


51 
50 

49 


9.80 957 


9.92 638 


0.07 362 


9.88 319 


9.80 972 


9.92 663 


0.07 337 


9.88 308 


12 


9.80 987 


9.92 689 


0.07 311 


9.88 298 


48 


13 


9.81 002 


9.92 715 


0.07 285 


9.88 287 


47 


14 


9.81 017 


9.92 740 


0.07 260 


9.88 276 


46 


15 


9.81 032 


9.92 766 


0.07 234 


9.88 266 


45 


16 


9.81 047 


9.92 792 


0.07 208 


9.88 255 


44 


17 


9.81 061 


9.92 817 


0.07 183 


9.88 244 


43 


18 


9.81 076 


9.92 843 


0.07 157 


9.88 234 


42 


19 
20 

21 


9.81 091 


9.92 868 


0.07 132 


9.88 223 


41 
40 

39 


9.81 106 


9.92 894 


0.07 106 


9.88 212 


9.81 121 


9.92 920 


0.07 080 


9.88 201 


22 


9.81 136 


9.92 945 


0.07 055 


9.88 191 


38 


23 


9.81 151 


9.92 971 


0.07 029 


9.88 180 


37 


24 


9.81 166 


9.92 996 


0.07 004 


9.88 169 


36 


25 


9.81 180 


9.93 022 


0.06 978 


9.88 158 


35 


26 


9.81 195 


9.93 048 


0.06 952 


9.88 148 


34 


27 


9.81 210 


9.93 073 


0.06 927 


9.88 137 


33 


28 


9.81 225 


9.93 099 


0.06 901 


9.88 126 


32 


29 - 

30 

31 


9.81 240 


9.93 124 


0.06 876 


9.88 115 


31 
30 

29 


9.81 254 


9.93 150 


0.06 850 


9.88 105 


9.81 269 


9.93 175 


0.06 825 


9.88 094 


32 


9.81 284 


9.93 201 


0.06 799 


9.88 083 


28 


33 


9.81 299 


9.93 227 


0.06 773 


9.88 072 


27 


34 


9.81 314 


9.93 252 


0.06 748 


9.88 061 


26 


35 


9.81 328 


9.93 278 


0.06 722 


9.88 051 


25 


36 


9.81 343 


9.93 303 


0.06 697 


9.88 040 


24 


37 


9.81 358 


9.93 329 


0.06 671 


9.88 029 


23 


38 


9.81 372 


9.93 354 


0.06 646 


9.88 018 


22 


39 
40 
41 


9.81 387 


9.93 380 


0.06 620 


9.88 007 


21 
20 

19 


9.81 402 


9.93 406 


0.06 594 


9.87 996 


9.81 417 


9.93 431 


0.06 569 


9.87 985 


42 


9.81 431 


9.93 457 


0.06 543 


9.87 975 


18 


43 


9.81 446 


9.93 482 


0.06 518 


9.87 964 


17 


44 


9.81 461 


9.93 508 


0.06 492 


9.87 953 


16 


45 


9.81 475 


9.93 533 


0.06 467 


9.87 942 


15 


46 


9.81 490 


9.93 559 


0.06 441 


9.87 931 


14 


47 


9.81 505 


9.93 584 


0.06 416 


9.87 920 


13 


48 


9.81 519 


9.93 610 


0.06 390 


9.87 909 


12 


49 
50 

51 


9.81 534 


9.93 636 


0.06 364 


9.87 898 


11 
10 

9 


9.81 549 


9.93 661 


0.06 339 


9.87 887 


9.81 563 


9.93 687 


0.06 313 


9.87 877 


52 


9.81 578 


9.93 712 


0.06 288 


9.87 866 


8 


53 


9.81 592 


9.93 738 


0.06 262 


9.87 855 


7 


54 


9.81 607 


9.93 763 


0.06 237 


9.87 844 


6 


55 


9.81 622 


9.93 789 


0.06 211 


9.87 833 


5 


56 


9.81 636 


9.93 814 


0.06 186 


9.87 822 


4 


57 


9.81 651 


9.93 840 


0.06 160 


9.87 811 


3 


58 


9.81 665 


9.93 865 


0.06 135 


9.87 800 


2 


59 
60 


9.81 680 


9.93 891 


0.06 109 


9.87 789 


1 



9.81 694 


9.93 916 


0.06 084 


9.87 778 




L. Cos. 


L. Cot. 


L. Tan. 


L. Sin. 


9 






4! 


9° 







68 



41' 



t 


L Sin. 


L. Tan. 


L. Cot. 


L. Cos. 






1 


9.81 694 


9.93 916 


0.06 084 


9.87 778 


60 

59 


9.81 709 


9.93 942 


0.06 058 


9.87 767 


2 


9.81 723 


9.93 967 


0.06 033 


9.87 756 


58 


3 


9.81 738 


9.93 993 


0.06 007 


9.87 745 


57 


4 


9.81 752 


9.94 018 


0.05 982 


9.87 734 


56 


5 


9.81 767 


9.94 044 


0.05 956 


9.87 723 


55 


6 


9.81 781 


9.94 069 


0.05 931 


9.87 712 


54 


7 


9.81 796 


9.94 095 


0.05 905 


9.87 701 


53 


8 


9.81 810 


9.94 120 


0.05 880 


9.87 690 


52 


9 
10 

11 


9.81 825 


9.94 146 


0.05 854 


9.87 679 


51 
50 

49 


9.81 839 


9.94 171 


0.05 829 


9.87 668 


9.81 854 


9.94 197 


0.05 803 


9.87 657 


12 


9.81 868 


9.94 222 


0.05 778 


9.87 646 


48 


13 


9.81 882 


9.94 248 


0.05 752 


9.87 635 


47 


14 


9.81 897 


9.94 273 


0.05 727 


9.87 624 


46 


15 


9.81 911 


9.94 299 


0.05 701 


9.87 613 


45 


16 


9.81 926 


9.94 324 


0.05 676 


9.87 601 


44 


17 


9.81 940 


9.94 350 


0.05 650 


9.87 590 


43 


18 


9.81 955 


9.94 375 


0.05 625 


9.87 579 


42 


19 
20 

21 


9.81 969 


9.94 401 


0.05 599 


9.87 568 


41 
40 
39 


9.81 983 


9.94 426 


0.05 574 


9.87 557 


9.81 998 


• 9.94 452 


0.05 548 


9.87 546 


22 


9.82 012 


9.94 477 


0.05 523 


9.87 535 


38 


23 


9.82 026 


9.94 503 


0.05 497 


9.87 524 


37 


24 


9.82 041 


9.94 528 


0.05 472 


9.87 513 


36 


25 


9.82 055 


9.94 554 


0.05 446 


9.87 501 


35 


26 


9.82 069 


9.94 579 


0.05 421 


9.87 490 


34 


27 


9.82 084 


9.94 604 


0.05 396 


9.87 479 


33 


28 


9.82 098 


9.94 630 


0.05 370 


9.87 468 


32 


29 
30 

31 


9.82 112 


9.94 655 


0.05 345 


9.87 457 


31 
30 

29 


9.82 126 


9.94 681 


0.05 319 


9.87 446 


9.82 141 


9.94 706 


0.05 294 


9.87 434 


32 


9.82 155 


9.94 732 


0.05 268 


9.87 423 


28 


33 


9.82 169 


9.94 757 


0.05 243 


9.87 412 


27 


34 


9.82 184 


9.94 783 


0.05 217 


9.87 401 


26 


35 


9.82 198 


9.94 808 


0.05 192 


9.87 390 


25 


36 


9.82 212 


9.94 834 


0.05 166 


9.87 378 


24 


37 


9.82 226 


9.94 859 


0.05 141 


9.87 367 


23 


38 


9.82 240 


9.94 884 


0.05 116 


9.87 356 


22 


39 
40 
41 


9.82 255 


9.94 910 


0.05 090 


9.87 345 


21 
20 

19 


9.82 269 


9.94 935 


0.05 063 


9.87 334 


9.82 283 


9.94 961 


0.05 039 


9.87 322 


42 


9.82 297 


9.94 986 


0.05 014 


9.87 311 


18 


43 


9.82 311 


9.95 012 


0.04 988 


9.87 300 


17 


44 


9.82 326 


9.95 037 


0.04 963 


9.87 288 


16 


45 


9.82 340 


9.95 062 


0.04 938 


9.87 277 


15 


46 


9.82 354 


9.95 088 


0.04 912 


9.87 266 


14 


47 


9.82 368 


9.95 113 


0.04 887 


9.87 255 


13 


48 


9.82 382 


9.95 139 


0.04 861 


9.87 243 


12 


49 
50 
51 


9.82 396 


9.95 164 


0.04 836 


9.87 232 


11 
10 

9 


9.82 410 


9.95 190 


0.04 810 


9.87 221 


9.82 424 


9.95 215 


0.04 785 


9.87 209 


52 


9.82 439 


9.95 240 


0.04 760 


9.87 198 


8 


53 


9.82 453 


9.95 266 


0.04 734 


9.87 187 


7 


54 


9.82 467 


9.95 291 


0.04 709 


9.87 175 


6 


55 


9.82 481 


9.95 317 


0.04 683 


9.87 164 


5 


56 


9.82 495 


9.95 342 


0.04 658 


9.87 153 


4 


57 


9.82 509 


9.95 368 


0.04 632 


9.87 141 


3 


58 


9.82 523 


9.95 393 


0.04 007 


9.87 130 


2 


59 
60 


9.82 537 


9.95 418 


0.04 582 


9.87 119 


1 



9.82 551 


9.95 444 


0.04 556 


9.87 107 




L. Cos. 


L. Cot. 


L. Tan. 


L. Sin. 


9 



48' 



42' 



69 



t 


L. Sin. 


L.Tan. 


L. Cot. 


L. Cos. 






1 


9.82 551 


9.95 444 


0.04 556 


9.87 107 


60 

59 


9.82 565 


9.95 469 


0.04 531 


9.87 096 


2 


9.82 579 


9.95 495 


0.04 505 


9.87 085 


58 


3 


9.82 593 


9.95 520 


0.04 480 


9.87 073 


57 


4 


9.82 607 


9.95 545 


0.04 455 


9.87 062 


56 


5 


9.82 621 


9.95 571 


0.04 429 


9.87 050 


55 


6 


9.82 635 


9.95 596 


0.04 404 


9.87 039 


54 


7 


9.82 649 


9.95 622 


0.04 378 


9.87 028 


53 


8 


9.82 663 


9.95 647 


0.04 353 


9.87 016 


52 


9 
10 

11 


9.82 677 


9.95 672 


0.04 328 


9.87 005 


51 
50 

49 


9.82 691 


9.95 698 


0.04 302 


9.86 993 


9.82 705 


9.95 723 


0.04 277 


9.86 982 


12 


9.82 719 


9.95 748 


0.04 252 


9.86 970 


48 


13 


9.82 733 


9.95 774 


0.04 226 


9.86 959 


47 


14 


9.82 747 


9.95 799 


0.04 201 


9.86 947 


46 


15 


9.82 761 


9.95 825 


0.04 175 


9.86 936 


45 


16 


9.82 775 


9.95 850 


0.04 150 


9.86 924 


44 


17 


9.82 788 


9.95 875 


0.04 125 


9.86 913 


43 


18 


9.82 802 


9.95 901 


0.04 099 


9.86 902 


42 


19 
20 

21 


9.82 816 


9.95 926 


0.04 074 


9.86 890 


41 
40 

39 


9.82 830 


9.95 952 


0.04 048 


9.86 879 


9.82 844 


9.95 977 


0.04 023 


9.86 867 


22 


9.82 858 


9.96 002 


0.03 998 


9.86 855 


38 


23 


9.82 872 


9.96 028 


0.03 972 


9.86 844 


37 


24 


9.82 885 


9.96 053 


0.03 947 


9.86 832 


36 


25 


9.82 899 


9.96 078 


0.03 922 


9.86 821 


35 


26 


9.82 913 


9.96 104 


0.03 896 


9.86 809 


34 


27 


9.82 927 


9.96 129 


0.03 871 


9.86 798 


33 


28 


9.82 941 


9.96 155 


0.03 845 


9.86 786 


32 


29 
30 

31 


9.82 955 


9.96 180 


0.03 820 


9.86 775 


31 
30 

29 


9.82 968 


9.96 205 


0.03 795 


9.86 763 


9.82 982 


9.96 231 


0.03 769 


9.86 752 


32 


9.82 996 


9.96 256 


0.03 744 


9.86 740 


28 


33 


9.83 010 


9.96 281 


0.03 719 


9.86 728 


27 


34 


9.83 023 


9.96 307 


0.03 693 


9.86 717 


26 


35 


9.83 037 


9.96 332 


0.03 668 


9.86 705 


25 


36 


9.83 051 


9.96 357 


0.03 643 


9.86 694 


24 


37 


9.83 065 


9.96 383 


0.03 617 


9.86 682 


23 


38 


9.83 078 


9.96 408 


0.03 592 


9.86 670 


22 


39 
40 

41 


9.83 092 


9.96 433 


0.03 567 


9.86 659 


21 
20 

19 


9.83 106 


9. 96 459 


0.03 541 


9.86 647 


9.83 120 


9.96 484 


0.03 516 


9.86 635 


42 


9.83 133 


9.96 510 


0.03 490 


9.86 624 


18 


43 


9.83 147 


9.96 535 


0.03 465 


9.86 612 


17 


44 


9.83 161 


9.96 560 


0.03 440 


9.86 600 


16 


45 


9.83 174 


9.96 586 


0.03 414 


9.86 589 


15 


46 


9.83 188 


9.96 611 


0.03 389 


9.86 577 


14 


47 


9.83 202 


9.96 636 


0.03 364 


9.86 565 


13 


48 


9.83 215 


9.96 662 


0.03 338 


9.86 554 


12 


49 
50 

51 


9.83 229 


9.96 687 


0.03 313 


9.86 542 


11 
10 

9 


9.83 242 


9.96 712 


0.03 288 


9.86 530 


9.83 256 


9.96 738 


0.03 262 


9.86 518 


52 


9.83 270 


9.96 763 


0.03 237 


9.86 507 


8 


53 


9.83 283 


9.96 788 


0.03 212 


9.86 495 


7 


54 


9.83 297 


9.96 814 


0.03 186 


9.86 483 


6 


55 


9.83 310 


9.96 839 


0.03 161 


9.86 472 


5 


56 


9.83 324 


9.96 864 


0.03 136 


9.86 460 


4 


57 


9.83 338 


9.96 890 


0.03 110 


9.86 448 


3 


58 


9.83 351 


9.96 915 


0.03 085 


9.86 436 


2 


59 
60 


9.83 365 


9.96 940 


0.03 060 


9.86 42§ 


1 



9.83 378 


9.96 966 


0.03 034 


9.86 413 




L. Cos. 


L. Cot. 


L. Tan. 


L. Sin. 


9 



47' 



7o 



43' 





» 


L. Sin. 


L. Tan. 


L. Cot. 


L. Cos. 








1 


9.83 378 


9.96 966 


0.03 034 


9.86 413 


60 

59 




9.83 392 


9.96 991 


0.03 009 


9.86 401 




2 


9.83 405 


9.97 016 


0.02 984 


9.86 389 


58 




3 


9.83 419 


9.97 042 


0.02 958 


9.86 377 


57 




4 


9.83 432 


9.97 067 


0.02 933 


9.86 366 


56 




5 


9.83 446 


9.97 092 


0.02 908 


9.86 354 


55 




6 


9.83 459 


9.97 118 


0.02 882 


9.86 342 


54 




7 


9.83 473 


9.97 143 


0.02 857 


9.86 330 


53 




8 


9.83 486 


9.97 168 


0.02 832 


9.86 318 


52 




9 
10 

11 


9.83 500 


9.97 193 


0.02 807 


9.86 306 


51 
50 

49 




9.83 513 


9.97 219 


0.02 781 


9.86 295 




9.83 527 


9.97 244 


0.02 756 


9.86 283 




12 


9.83 540 


9.97 269 


0.02 731 


9.86 271 


48 




13 


9.83 554 


9.97 295 


0.02 705 


9.86 259 


47 




14 


9.83 567 


9.97 320 


0.02 680 


9.86 247 


46 




15 


9.83 581 


9.97 345 


0.02 655 


9.86 235 


45 




16 


9.83 594 


9.97 371 


0.02 629 


9.86 223 


44 




17 


9.83 608 


9.97 396 


0.02 604 


9.86 211 


43 




18 


9.83 621 


9.97 421 


0.02 579 


9.86 200 


42 




19 
20 
21 


9.83 634 


9.97 447 


0.02 553 


9.86 188 


41 
40 

39 




9.83 648 


9.97 472 


0.02 528 


9.86 176 




9.83 661 


9.97 497 


0.02 503 


9.86 164 




22 


9.83 674 


9.97 523 


0.02 477 


9.86 152 


38 




23 


9.83 688 


9.97 548 


0.02 452 


9.86 140 


37 




24 


9.83 701 


9.97 573 


0.02 427 


9.86 128 


36 




25 


9.83 715 


9.97 598 


0.02 402 


9.86 116 


35 




26 


9.83 728 


9.97 624 


0.02 376 


9.86 104 


34 




27 


9.83 741 


9.97 649 


0.02 351 


9.86 092 


33 




28 


9.83 755 


9.97 674 


0.02 326 


9.86 080 


32 




29 
30 

31 


9.83 768 


9.97 700 


0.02 300 


9.86 068 


31 
30 
29 




9.83 781 


9.97 725 


0.02 275 


9.86 056 




9.83 795 


9.97 750 


0.02 250 


9.86 044 




32 


9.83 808 


9.97 776 


0.02 224 


9.86 032 


28 




33 


9.83 821 


9.97 801 


0.02 199 


9.86 020 


27 




34 


9.83 834 


9.97 826 


0.02 174 


9.86 008 


26 




35 


9.83 848 


9.97 851 


0.02 149 


9.85 996 


25 




36 


9.83 861 


9.97 877 


0.02 123 


9.85 984 


24 




37 


9.83 874 


9.97 902 


0.02 098 


9.85 972 


23 




38 


9.83 887 


9.97 927 


0.02 073 


9.85 960 


22 




39 
40 

41 


9.83 901 


9.97 953 


0.02 047 


9.85 948 


21 
20 

19 




9.83 914 


9.97 978 


0.02 022 


9.85 936 




9.83 927 


9.98 003 


0.01 997 


9.85 924 




42 


9.83 940 


9.98 029 


0.01 971 


9.85 912 


18 




43 


9.83 954 


9.98 054 


0.01 946 


9.85 900 


17 




44 


9.83 967 


9.98 079 


0.01 921 


9.85 888 


16 




45 


9.83 980 


9.98 104 


0.01 896 


9.85 876 


15 




46 


9.83 993 


9.98 130 


0.01 870 


9.85 864 


14 




47 


9.84 006 


9.98 155 


0.01 845 


9.85 851 


13 




48 


9.84 020 


9.98 180 


0.01 820 


9.85 839 


12 




49 
50 

51 


9.84 033 


9.98 206 


0.01 794 


9.85 827 


11 
10 

9 




9.84 046 


9.98 231 


0.01 769 


9.85 815 




9.84 059 


9.98 256 


0.01 744 


9.85 803 




52 


9.84 072 


9.98 281 


0.01 719 


9.85 791 


8 




53 


9.84 085 


9.98 307 


0.01 693 


9.85 779 


7 




54 


9.84 098 


9.98 332 


0.01 668 


9.85 766 


6 




55 


9.84 112 


9.98 357 


0.01 643 


9.85 754 


5 




56 


9.84 125 


9.98 383 


0.01 617 


9.85 742 


4 




57 


9.84 138 


9.98 408 


0.01 592 


9.85 730 


3 




58 


9.84 151 


9.98 433 


0.01 567 


9.85 718 


2 




59 
60 


9.84 164 


9.98 458 


0.01 542 


9.85 706 


1 





9.84 177 


9.98 484 


0.01 516 


9.85 693 




L. COS. 


L. Cot. 


L. Tan. 


L. Sin. 


t 








4 


*>o 







44' 



7i 



9 


Lo Sin. 


L. Tan. 


L. Cot. 


L. Cos. 






1 


9.84 177 


9.98 484- 


0.01 516 


9.85 693 


60 

59 


9.84 190 


9.98 509 


0.01 491 


9.85 681 


2 


9.81 203 


9.98 534 


0.01 466 


9.85 669 


58 


3 


9.84 216 


9.98 560 


0.01 440 


9.85 657 


57 


4 


9.81 229 


9.98 585 


0.01 415 


9.85 645 


56 


5 


9.84 242 


9.98 610 


0.01 390 


9.85 632 


55 


6 


9.84 255 


9.98 635 


0.01 365 


9.85 620 


54 


7 


9.84 269 


9.98 661 


0.01 339 


9.85 608 


53 


8 


9.84 282 


9.98 686 


0.01 314 


9.85 596 


52 


9 
10 

11 


9.84 295 


9.98 711 


0.01 289 


9.85 583 


51 
50 

49 


9.84 308 


9.98 737 


0.01 263 


9.85 571 


9.84 321 


9.98 762 


0.01 238 


9.85 559 


12 


9.84 334 


9.98 787 


0.01 213 


9.85 547 


48 


13 


9.84 347 


9.98 812 


0.01 188 


9.85 534 


47 


14 


9.84 360 


9.98 838 


0.01 162 


9.85 522 


46 


15 


9.84 373 


9.98 863 


0.01 137 


9.85 510 


45 


16 


9.84 385 


9.98 888 


0.01 112 


9.85 497 


44 


17 


9.84 398 


9.98 913 


0.01 087 


9.85 485 


43 


18 


9.84 411 


9.98 939 


0.01 061 


9.85 473 


42 


19 
20 

21 


9.84 424 


9.98 964 


0.01 036 


9.85 460 


41 
40 

39 


9.84 437 


9.98 989 


0.01 Oil 


9.85 448 


9.84 450 


9.99 015 


0.00 985 


9.85 436 


22 


9.84 463 


9.99 040 


0.00 960 


9.85 423 


38 


23 


9.84 476 


9.99 065 


0.00 935 


9.85 411 


37 


24 


9.84 489 


9.99 090 


0.00 910 


9.85 399 


36 


25 


9.84 502 


9.99 116 


0.00 884 


9.85 386 


35 


26 


9.84 515 


9.99 141 


0.00 859 


9.85 374 


34 


27 


9.84 528 


9.99 166 


0.00 834 


9.85 361 


33 


28 


9.84 540 


9.99 191 


0.00 809 


9.85 349 


32 


29 
80 

31 


9.84 553 


9.99 217 


0.00 783 


9.85 337 


31 
30 

29 


9.84 566 


9.99 242 


0.00 758 


9.85 324 


9.84 579 


9.99 267 


0.00 733 


9.85 312 


32 


9.84 592 


9.99 293 


0.00 707 


9.85 299 


28 


33 


9.84 605 


9.99 318 


0.00 682 


9.85 287 


27 


34 


9.84 618 


9.99 343 


0.00 657 


9.85 274 


26 


35 


9.84 630 


9.99 368 


0.00 632 


9.85 262 


25 


36 


9.84 643 


9.99 394 


0.00 606 


9.85 250 


24 


37 


9.84 656 


9.99 419 


0.00 581 


9.85 237 


23 


38 


9.84 669 


9.99 444 


0.00 556 


9.85 225 


22 


39 
40 

41 


9.84 682 


9.99 469 


0.00 531 


9.85 212 


21 
20 

19 


9.84 694 


9.99 495 


0.00 505 


9.85 200 


9.84 707 


9.99 520 


0.00 480 


9.85 187 


42 


9.84 720 


9.99 545 


0.00 455 


9.85 175 


18 


43 


9.84 733 


9.99 570 


0.00 430 


9.85 162 


17 


44 


9.84 745 


9.99 596 


0.00 404 


9.85 150 


16 


45 


9.84 758 


9.99 621 


0.00 379 


9.85 137 


15 


46 


9.84 771 


9.99 646 


0.00 354 


9.85 125 


14 


47 


9.84 784 


9.99 672, 


0.00 328 


9.85 112 


13 


48 


9.84 796 


9.99 697 


0.00 303 


9.85 100 


12 


49 
50 

51 


9.84 809 


9.99 722 


0.00 278 


9.85 087 


11 
10 

9 


9.84 822 


9.99 747 


0.00 253 


9.85 074 


9.84 835 


9.99 773 


0.00 227 


9.85 062 


52 


9.84 847 


9.99 798 


0.00 202 


9.85 049 


8 


53 


9.84 860 


9.99 823 


0.00 177 


9.85 037 


7 


54 


9.84 873 


9.99 848 


0.00 152 


9.85 024 


6 


55 


9.84 885 


9.99 874 


0.00 126 


9.85 012 


5 


56 


9.84 898 


9.99 899 


0.00 101 


9.84 999 


4 


57 


9.84 911 


9.99 924 


0.00 076 


9.84 986 


3 


58 


9.84 923 


9.99 949 


0.00 051 


9.84 974 


2 


59 
60 


9.84 936 


9.99 975 


0.00 025 


9.84 961 


1 



9.84 949 


10.00 000 


0.00 000 


9.84 949 




L. Cos. 


L. Cot. 


L. Tan. 


L. Sin. 


r 



45 



7 2 



S AND T TABLE 



TABLE III 

S and T Table 

to be used when the angle is between 0° and 2 C 
or between 88° and 90°. 

Formulas 
log sin x = log x (in seconds) + S 
log tan x = log x (in seconds) + T 



Angle 


x" 


log sin x 


S 


log tan x 


T 


0°00' 





OO 


4.68557 


— oo 


4.68557 


0°03' 


180 


6.94085 


4.68557 


6.94085 


4.68557 


0°04' 


240 


7.06579 


4.68557 


7.06579 


4.68558 


0°28' 


1680 


7.91088 


4.68557 


7.91089 


4.68558 


0°29' 


1740 


7.92612 


4.68557 


7.92613 


4.68559 


0°40' 


2400 


8.06578 


4.68557 


8.06581 


4.68559 


0°41' 


2460 


8.07650 


4.68556 


8.07653 


4.68560 


0°49' 


2940 


8.15391 


4.68556 


8.15395 


4.68560 


0°50' 


3000 


8.16268 


4.68556 


8.16273 


4.68561 


0°56' 


3360 


8.21189 


4.68556 


8.21195 


4.68561 


0°57' 


3420 


8.21958 


4.68555 


8.21964 


4.68561 


0°58' 


3480 


8.22713 


4.68555 


8.22720 


4.68562 


1°03' 


3780 


8.26304 


4.68555 


8.26312 


4.68562 


1°04' 


3840 


8.26988 


4.68555 


8.26996 


4.68563 


1°09' 


4140 


8.30255 


4.68555 


8.30263 


4.68563 


1°10' 


4200 


8.30879 


4.68554 


8.30888 


4.68563 


1°11' 


4260 


8.31495 


4.68554 


8.31505 


4.68564 


1°15' 


4500 


8.33875 


4.68554 


8.33886 


4.68564 


1°16' 


4560 


8.34450 


4.68554 


8.34461 


4.68565 


1°20' 


4800 


8.36678 


4.68554 


8.36689 


4.68565 i 


1°21' 


4860 


8.37217 


4.68553 


8.37229 


4.68566 


1°25' 


5100 


8.39310 


4.68553 


8.39323 


4.68566 


1°26' 


5160 


8.39818 


4.68553 


8.39832 


4.68567 


1°30' 


5400 


8.41792 


4.68553 


8.41807 


4.68567 


1°31' 


5460 


8.42272 


4.68552 


8.42287 


4.68568 


1°34' 


5640 


8.43680 


4.68552 


8.43696 


4.68568 


1°35' 


5700 


8.44139 


4.68552 


8.44156 


4.68569 


1°38' 


5880 


8.45489 


4.68552 


8.45507 


4.68569 


1°39' 


5940 


8.45930 


4.68551 


8.45948 


4.68569 


1°40' 


6000 


8.46366 


4.68551 


8.46385 


4.68570 


1°43' 


6180 


8.47650 


4.68551 


8.47669 


4.68570 


1°44' 


6240 


8.48069 


4.68551 


8.48089 


4.68571 


1°46' 


6360 


8.48896 


4.68551 


8.48917 


4.68571 


jo 47 ' 


6420 


8.49304 


4.68550 


8.49325 


4.68572 


1°50' 


6600 


8.50504 


4.68550 


8.50527 


4.68572 


1°51' 


6660 


8.50897 


4.68550 


8.50920 


4.68573 


1°54' 


6840 


8.52055 


4.68550 


8.52079 


4.68573 


1°55' 


6900 


8.52434 


4.68549 


8.52459 


4.68574 


1°57' 


7020 


8.53183 


4.68549 


8.53208 


4.68574 


1°58' 


7080 


8.53552 


4.68549 


"8.53578 


4.68575 


2° 00' 


7200 


8.54282 


~ 4.68549 


8.54308 


~ 4.68575 



TABLE IV. — NATURAL FUNCTIONS 



73 



1 


G 


° 


1° 


2° 


3 


o 


4 


o 


/ 




1 

2 
3 
4 

5 

6 

7 
8 
9 

10 

11 

12 
13 
14 

15 

16 
17 
18 
19 

20 

21 
22 
23 
24 

25 

26 
27 
28 
29 

30 

31 
32 
33 
34 

35 

36 
37 
38 
39 

40 

41 
42 
43 
44 

45 

46 
47 
48 
49 

50 

51 
52 
53 
54 

55 

56 
57 
58 
59 

60 


sin 


cos 


sin 


cos 


sin 


COS 


sin 


COS 


sin 


cos 


60 

59 
58 
57 
56 

55 

54 
53 
52 
51 

50 

49 
48 
47 
46 

45 

44 
43 
42 
41 

40 

39 
38 
37 
36 

35 

34 
33 
32 
31 
30 
29 
28 
27 
26 

25 

24 
23 
22 
21 

20 

19 
18 
17 
16 

15 

14 
13 

12 
11 

10 

9 

8 
7 
6 

5 

4 
3 
2 
1 




0000 


1.000 


0175 


9998 


0349 


9994 


0523 


9986 


0698 


9976 


0003 
0006 
0009 
0012 


1.000 
1.000 
1.000 
1.000 


0177 
0180 
0183 
0186 


9998 
9998 
9998 
9998 


0352 
0355 
0358 
0361 


9994 
9994 
9994 
9993 


0526 
0529 
0532 
0535 


9986 
9986 
9986 
9986 


0700 
0703 
0706 
0709 


9975 
9975 
9975 
9975 


0015 


1.000 


0189 


9998 


0364 


9993 


0538 


9986 


0712 


9975 


0017 
0020 
0023 
0026 


1.000 
1.000 
1.000 
1.000 


0192 
0195 
0198 
0201 


9998 
9998 
9998 
9998 


0366 
0369 
0372 
0375 


9993 
9993 
9993 
9993 


0541 
0544 
0547 
0550 


9985 
9985 
9985 
9985 


0715 
0718 
0721 
0724 


9974 
9974 
9974 
9974 


0029 


1.000 


0204 


9998 


0378 


9993 


0552 


9985 


0727 


9974 


0032 
0035 
0038 
0041 


1.000 
1.000 
1.000 
1.000 


0207 
0209 
0212 
0215 


9998 
9998 
9998 
9998 


0381 
0384 
0387 
0390 


9993 
9993 
9993 
9992 


0555 
0558 
0561 
0564 


9985 
9984 
9984 
9984 


0729 
0732 
0735 
0738 


9973 
9973 
9973 
9973 


0044 


1.000 


0218 


9998 


0393 


9992 


0567 


9984 


0741 


9973 


0047 
0049 
0052 
0055 


1.000 
1.000 
1.000 
1.000 


0221 
0224 
0227 
0230 


9998 
9997 
9997 
9997 


0396 
0398 
0401 
0404 


9992 
9992 
9992 
9992 


0570 
0573 
0576 
0579 


9984 
9984 
9983 
9983 


0744 
0747 
0750 
0753 


9972 
9972 
9972 
9972 


0058 


1.000 


0233 


9997 


0407 


9992 


0581 


9983 


0756 


9971 


0061 
0064 
0067 
0070 


1.000 
1.000 
1.000 
1.000 


0236 
0239 
0241 
0244 


9997 
9997 
9997 
9997 


0410 
0413 
0416 
0419 


9992 
9991 
9991 
9991 


0584 
0587 
0590 
0593 


9983 
9983 
9983 
9982 


0758 
0761 
0764 
0767 


9971 
9971 
9971 
9971 


0073 


1.000 


0247 


9997 


0422 


9991 


0596 


9982 


0770 


9970 


0076 
0079 
0081 
0084 


1.000 
1.000 
1.000 
1.000 


0250 
0253 
0256 
0259 


9997 
9997 
9997 
9997 


0425 
0427 
0430 
0433 


9991 
9991 
9991 
9991 


0599 
0602 
0605 
0608 


9982 
9982 
9982 
9982 


0773 
0776 
0779 
0782 


9970 
9970 
9970 
9969 


0087 


1.000 


0262 


9997 


0436 


9990 


0610 


9981 


0785 


9969 


0090 
0093 
0096 
0099 


1.000 
1.000 
1.000 
1.000 


0265 
0268 
0270 
0273 


9996 
9996 
9996 
9996 


0439 
0442 
0445 
0448 


9990 
9990 
9990 
9990 


0613 
0616 
0619 
0622 


9981 
9981 
9981 
9981 


0787 
0790 
0793 
0796 


9969 
9969 
9968 
9968 


0102 


9999 


0276 


9996 


0451 


9990 


0625 


9980 


0799 


9968 


0105 
0108 
0111 
0113 


9999 
9999 
9999 
9999 


0279 
0282 
0285 
0288 


9996 
9996 
9996 
9996 


0454 
0457 
0459 
0462 


9990 
9990 
9989 
9989 


0628 
0631 
0634 
0637 


9980 
9980 
9980 
9980 


0802 
0805 
0808 
0811 


9968 
9968 
9967 
9967 


0116 


9999 


0291 


9996 


0465 


9989 


0640 


9980 


0814 


9967 


0119 
0122 
0125 
0128 


9999 
9999 
9999 
9999 


0294 
0297 
0300 
0302 


9996 
9996 
9996 
9995 


0468 
0471 
0474 
0477 


9989 
9989 
9989 
9989 


0642 
0645 
0648 
0651 


9979 
9979 
9979 
9979 


0816 
0819 
0822 
0825 


9967 
9966 
9966 
9966 


0131 


9999 


0305 


9995 


0480 


9988 


0654 


9979 


0828 


9966 


0134 
0137 
0140 
0143 


9999 
9999 
9999 
9999 


0308 
0311 
0314 
0317 


9995 
9995 
9995 
9995 


0483 
0486 
0488 
0491 


9988 
9988 
9988 
9988 


0657 
0660 
0663 
0666 


9978 
9978 
9978 
9978 


0831 
0834 
0837 
0840 


9965 
9965 
9965 
9965 


0145 


9999 


0320 


9995 


0494 


9988 


0669 


9978 


0843 


9964 


0148 
0151 
0154 
0157 


9999 
9999 
9999 
9999 


0323 
0326 
0329 
0332 


9995 
9995 
9995 
9995 


0497 
0500 
0503 
0506 


9988 
9987 
9987 
9987 


0671 
0674 
0677 
0680 


9977 
9977 
9977 
9977 


0845 
0848 
0851 
0854 


9964 
9964 
9964 
9963 


0160 


9999 


0334 


9994 


0509 


9987 


0683 


9977 


0857 


9963 


0163 
0166 
0169 
0172 


9999 
9999 
9999 
9999 


0337 
0340 
0343 
0346 


9994 
9994 
9994 
9994 


0512 
0515 
0518 
0520 


9987 
9987 
9987 
9986 


0686 
0689 
0692 
0695 


9976 
9976 
9976 
9976 


0860 
0863 
0866 
0869 


9963 
9963 
9962 
9962 


0175 


9999 


0349 


9994 


0523 


9986 


0698 


9976 


0872 


9962 


COS 


sin 


cos 


sin 


COS 


sin 


COS 


sin 


cos 


sin 


/ 


89° 


88° 


87° 


86° 


85° 


/ 



74 



NATURAL SINES AND COSINES 





1 


5 


o 


e 


o 


>y O 


8° 


9° 


/ 






1 

2 
3 

4 

5 

6 

7 
8 
9 

10 

11 
12 
13 
14 

15 

16 
17 
18 
19 

30 

21 
22 
23 
24 

35 

26 
27 
28 
29 

30 

31 
32 

33 
34 

35 

36 
37 
38 
39 

40 

41 

42 
43 
44 

45 

46 

47 

48 

■ 49 

50 

51 
52 
53 

54 

55 

56 
57 
58 
59 

60 


sin 


COS 


sin 


COS 


sin 


cos 


sin 


cos 


sin 


cos 


60 

59 
58 
57 
56 

55 

54 
53 
52 
51 

50 

49 
48 
47 
46 

45 

44 
43 
42 
41 

40 

39 
38 
37 
36 

35 

34 
33 
32 
31 
30 
29 
28 
27 
26 

35 

24 
23 
22 
21 

30 

19 
18 
17 
16 

15 

14 
13 
12 
11 

10 

9 
8 
7 
6 

5 

4 
3 
2 
1 






0872 


9962 


1045 


9945 


1219 


9925 


1392 


9903 


1564 


9877 




0874 
0877 
0880 
0883 


9962 
9961 
9961 
9961 


1048 
1051 
1054 
1057 


9945 
9945 
9944 
9944 


1222 
1224 
1227 
1230 


9925 
9925 
9924 
9924 


1395 
1397 
1400 
1403 


9902 
9902 
9901 
9901 


1567 
1570 
1573 
1576 


9876 
9876 
9876 
9875 




0886 


9961 


1060 


9944 


1233 


9924 


1406 


9901 


1579 


9875 




0889 
0892 
0895 
0898 


9960 
9960 
9960 
9960 


1063 
1066 
1068 
1071 


9943 
9943 
9943 
9942 


1236 
1239 
1241 
1245 


9923 
9923 
9923 
9922 


1409 
1412 
1415 
1418 


9900 
9900 
9899 
9899 


1582 
1584 
1587 
1590 


9874 
9874 
9873 
9873 




0901 


9959 


1074 


9942 


1248 


9922 


1421 


9899 


1593 


9872 




0903 
0906 
0909 
0912 


9959 
9959 
9959 
9958 


1077 
1080 
1083 
1086 


9942 
9942 
9941 
9941 


1250 
1253 
1256 
1259 


9922 
9921 
9921 
9920 


1423 
1426 
1429 
1432 


9898 
9898 
9897 
9897 


1596 
1599 
1602 
1605 


9872 
9871 
9871 
9870 




0915 


9958 


1089 


9941 


.1262 


9920 


1435 


9897 


1607 


9870 




0918 
0921 
0924 
0927 


9958 
9958 
9957 
9957 


1092 
1094 
1097 
1100 


9940 
9940 
9940 
9939 


1265 
1268 
1271 
1274 


9920 
9919 
9919 
9919 


1438 
1441 
1444 
1446 


9896 
9896 
9895 
9895 


1610 
1613 
1616 
1619 


9869 
9869 
9869 
9868 




0929 


9957 


1103 


9939 


1276 


9918 


1449 


9894 


1622 


9868 


1 


0932 
0935 
0938 
0941 


9956 
9956 
9956 
9956 


1106 
1109 
1112 
1115 


9939 
9938 
9938 
9938 


1279 
1282 
1285 
1288 


9918 
9917 
9917 
9917 


1452 
1455 
1458 
1461 


9894 
9894 
9893 
9893 


1625 
1628 
1630 
1633 


9867 
9867 
9866 
9866 


( 


0944 


9955 


1118 


9937 


1291 


9916 


1464 


9892 


1636 


9865 




■ 0947 
0950 
0953 
0956 


9955 
9955 
9955 
9954 


1120 
1123 
1126 
1129 


9937 
9937 
9936 
9936 


1294 
1297 
1299 
1302 


9916 
9916 
9915 
9915 


1467 
1469 
1472 
1475 


9892 
9891 
9891 
9891 


1639 
1642 
1645 
1648 


9865 
9864 
9864 
9863 




0958 


9954 


1132 


9936 


1305 


9914 


1478 


9890 


1650 


9863 




0961 
0964 
0967 
0970 


9954 
9953 
9953 
9953 


1135 
1138 
1141 
1144 


9935 
9935 
9935 
9934 


1308 
1311 
1314 
1317 


9914 
9914 
9913 
9913 


1481 
1484 
1487 
1490 


9890 
9889 
9889 
9888 


1653 
1656 
1659 
1662 


9862 
9862 
9861 
9861 




0973 


9953 


1146 


9934 


1320 


9913 


1492 


9888 


1665 


9860 




0976 
0979 
0982 
0985 


9952 
9952 
9952 
9951 


1149 
1152 
1155 
1158 


9934 
9933 
9933 
9933 


1323 
1325 
1328 
1331 


9912 
9912 
9911 
9911 


1495 
1498 
1501 
1504 


9888 
9887 
9887 
9886 


1668 
1671 
1673 
1676 


9860 
9859 
9859 
9859 




0987 


9951 


1161 


9932 


1334 


9911 


1507 


9886 


1679 


9858 




0990 
0993 
0996 
0999 


9951 
9951 
9950 
9950 , 


1164 
1167 
1170 
1172 


9932 
9932 
9931 
9931 


1337 
1340 
1343 
1346 


9910 
9910 
9909 
9909 


1510 
1513 
1515 
1518 


9885 
9885 
9884 
9884 


1682 
1685 
1688 
1691 


9858 
9857 
9857 
9856 




1002 


9950 


1175 


9931 


1349 


9909 


1521 


9884 


1693 


9856 




1005 
1008 
1011 
1013 


9949 
9949 
9949 
9949 


1178 
1181 
1184 
1187 


9930 
9930 
9930 
9929 


1351 
1354 
1357 
1360 


9908 
9908 
9907 
9907 


1524 
1527 
1530 
1533 


9883 
9883 
9882 
9882 


1696 
1699 

1702 
1705 


9855 
9855 
9854 
9854 




1016 


9948 


1190 


9929 


1363 


9907 


1536 


9881 


1708 


9853 




1019 
1022 
1025 
1028 


9948 
9948 
9947 
9947 


1193 
1196 
1198 
1201 


9929 
9928 
9928 
9928 


1366 
1369 
1372 
1374 


9906 
9906 
9905 
9905 


1538 
1541 
1544 
1547 


9881 
9880 
9880 
9880 


1711 
1714 
1716 
1719 


9853 
9852 
9852 
9851 




1031 


9947 


1204 


9927 


1377 


9905 


1550 


9879 


1722 


9851 




1034 
1037 
1039 
1042 


9946 
9946 
9946 
9946 


1207 
1210 
1213 
1216 


9927 
9927 
9926 
9926 


1380 
1383 
1386 
1389 


9904 
9904 
9903 
9903 


1553 
1556 
1559 
1561 


9879 
9878 
9878 
9877 


1725 
1728 
1731 
1734 


9850 
9850 
9849 
9849 




1045 


9945 


1219 


9925 


1392 


9903 


1564 


9877 


1736 


9848 




COS 


sin 


COS 


sin 


COS 


sin 


COS 


sin 


cos 


sin 






r 


84° 


83° 


83° 


81° 


80° 


1 



NATURAL SINES AND COSINES 



75 



1 


10° 


11° 


12° 


13° 


14° 


/ 




1 

2 
3 
4 

5 

6 

7 
8 
9 

10 

11 

12 
13 
14 

15 

16 
17 
18 
19 

20 

21 
22 
23 

24 

25 

26 
27 
28 
29 

30 

31 
32 
33 
34 

35 

36 
37 
38 

39 

40 

41 
42 
43 
44 

45 

46 
47 
48 
49 

50 

51 
52 
53 
54 

55 

56 
57 
58 
59 

60 


sin 


cos 


sin 


cos 


sin 


cos 


sin 


cos 


sin 


cos 


60 

59 
58 
57 
56 

55 

54 
53 

52 

51 

50 

49 
48 
47 
46 

45 

44 
43 
42 
41 

40 

39 
38 
37 
36 

35 

34 
33 
32 
31 
30 

29 
28 
27 
26 

25 

24 
23 
22 
21 

20 

19 
18 
17 
16 

15 

14 
13 
12 
11 

10 

9 
8 
7 
6 

5 

4 
3 
2 
1 



1736 


9848 


1908 


9816 


2079 


9781 


2250 


9744 


2419 


9703 


1739 
1742 
1745 
1748 


9848 
9847 
9847 
9846 


1911 
1914 
1917 
1920 


9816 
9815 
9815 
9814 


2082 
2085 
2088 
2090 


9781 
9780 
9780 
9779 


2252 
2255 
2258 
2261 


9743 
9742 
9742 
9741 


2422 
2425 
2428 
2431 


9702 
9702 
9701 
9700 


1751 


9846 


1922 


9813 


2093 


9778 


2264 


9740 


2433 


9699 


1754 
1757 
1759 
1762 


9845 
9845 
9844 
9843 


1925 
1928 
1931 
1934 


9813 
9812 
9812 
9811 


2096 
2099 
2102 
2105 


9778 
9777 
9777 
9776 


2267 
2269 
2272 
2275 


9740 
9739 
9738 
9738 


2436 
2439 
2442 
2445 


9699 
9698 
9697 
9697 


1765 


9843 


1937 


9811 


2108 


9775 


2278 


9737 


2447 


9696 


1768 
1771 
1774 
1777 


9842 
9842 
9841 
9841 


1939 
1942 
1945 
1948 


9810 
9810 
9809 
9808 


2110 
2113 
2116 
2119 


9775 
9774 
9774 
9773 


2281 
2284 
2286 
2289 


9736 
9736 
9735 
9734 


2450 
2453 
2456 
2459 


9695 
9694 
9694 
9693 


1779 


9840 


1951 


9808 


2122 


9772 


2292 


9734 


2462 


9692 


1782 
1785 
1788 
1791 


9840 
9839 
9839 
9838 


1954 
1957 
1959 
1962 


9807 
9807 
9806 
9806 


2125 
2127 
2130 
2133 


9772 
9771 
9770 
9770 


2295 
2298 
2300 
2303 


9733 
9732 
9732 
9731 


2464 
2467 
2470 
2473 


9692 
9691 
9690 
9689 


1794 


9838 


1965 


9805 


2136 


9769 


2306 


9730 


2476 


9689 


1797 
1799 
1802 
1805 


9837 
9837 
9836 
9836 


1968 
1971 
1974 
1977 


9804 
9804 
9803 
9803 


2139 
2142 
2145 
2147 


9769 
9768 
9767 
9767 


2309 
2312 
2315 
2317 


9730 
9729 
9728 
9728 


2478 
2481 
2484 
2487 


9688 
9687 
9687 
9686 


1808 


9835 


1979 


9802 


2150 


9766 


2320 


9727 


2490 


9685 


1811 
1814 
1817 
1819 


9835 
9834 
9834 
9833 


1982 
1985 
1988 
1991 


9802 
9801 
9800 
9800 


2153 
2156 
2159 
2162 


9765 
9765 
9764 
9764 


2323 
2326 
2329 
2332 


9726 
9726 
9725 
9724 


2493 
2495 
2498 
2501 


9684 
9684 
9683 
9682 


1822 


9833 


1994 


9799 


2164 


9763 


2334 


9724 


2504 


9681 


1825 
1828 
1831 
1834 


9832 
9831 
9831 
9830 


1997 
1999 
2002 
2005 


9799 
9798 
9798 
9797 


2167 
2170 
2173 
2176 


9762 
9762 
9761 
9760 


2337 
2340 
2343 
2346 


9723 
9722 
9722 
9721 


2507 
2509 
2512 
2515 


9681 
9680 
9679 
9679 


1837 


9830 


2008 


9796 


2179 


9760 


2349 


9720 


2518 


9678 


1840 
1842 
1845 
1848 


9829 
9829 
9828 
9828 


2011 
2014 
2016 
2019 


9796 
9795 
9795 
9794 


2181 
2184 
2187 
2190 


9759 
9759 
9758 
9757 


2351 
2354 
2357 
2360 


9720 
9719 
9718 
9718 


2521 
2524 
2526 
2529 


9677 
9676 
9676 
9675 


1851 


9827 


2022 


9793 


2193 


9757 


2363 


9717 


2532 


9674 


1854 
1857 
1860 
1862 


9827 
9826 
9826 
9825 


2025 
2028 
2031 
2034 


9793 
9792 
9792 
9791 


2196 
2198 
2201 
2204 


9756 
9755 
9755 
9754 


2366 
2368 
2371 
2374 


9716 
9715 
9715 
9714 


2535 
2538 
2540 
2543 


9673 
9673 
9672 
9671 


1865 


9825 


2036 


9790 


2207 


9753 


2377 


9713 


2546 


9670 


1868 
1871 
1874 
1877 


9824 
9823 
9823 
9822 


2039 
2042 
2045 
2048 


9790 
9789 
9789 
9788 


2210 
2213 
'2215 
2218 


9753 
9752 
9751 
9751 


2380 
2383 
2385 
2388 


9713 
9712 
9711 
9711 


2549 
2552 
2554 
2557 


9670 
9669 
9668 
9667 


1880 


9822 


2051 


9787 


2221 


9750 


2391 


9710 


2560 


9667 


1882 
1885 
1888 
1891 


9821 
9821 
9820 
9820 


2054 
2056 
2059 
2062 


9787 
9786 
9786 
9785 


2224 
2227 
2230 
2233 


9750 

9749 
9748 
9748 


2394 
2397 
2399 
2402 


9709 
9709 
9708 
9707 


2563 
2566 
2569 
2571 


9666 
9665 
9665 
9664 


1894 


9819 


2065 


9784 


2235 


9747 


2405 


9706 


2574 


9663 


1897 
1900 
1902 
1905 


9818 
9818 
9817 
9817 


2068 
2071 
2073 
2076 


9784 
9783 
9783 
9782 


2238 
2241 
2244 
2247 


9746 
9746 
9745 
9744 


2408 
2411 
2414 
2416 


9706 
9705 
9704 
9704 


2577 
2580 
2583 
2585 


9662 
9662 
9661 
9660 


1908 


9816 


2079 


9781 


2250 


9744 


2419 


9703 


2588 


9659 


COS 


sin 


COS 


sin 


cos 


sin 


COS 


sin 


COS 


sin 


/ 


79° 


78° 


77° 


76° 


75° 


/ 



76 



NATURAL SINES AND COSINES 



/ 


15° 


16° 


17° 


18° 


19° 


r 




1 

2 
3 

4 

5 

6 

7 
8 
9 

10 

11 
12 
13 

14 

15 

16 
17 
18 

19 

20 

21 

22 
23 
24 

25 

26 
27 

28 
29 

30 

31 
32 
33 
34 

35 

36 
37 
38 
39 

40 

41 
42 
43 
44 

45 

46 
47 
48 
49 

50 

51 
52 
53 

54 

55 

56 
57 
58 
59 

60 


sin 


cos 


sin 


cos 


sin 


COS 


sin 


cos 


sin 


cos 


60 

59 
58 
57 
56 

55 

54 
53 
52 
51 

50 

49 
48 
47 
46 
45 
44 
43 
42 
41 

40 

39 
38 
37 
36 

35 

34 
33 

32 
31 

30 

29 
28 
27 
26 

35 

24 
23 
22 
21 

20 

19 
18 
17 
16 

15 

14 
13 
12 
11 

10 

9 

8 
7 
6 

5 

4 
3 
2 
1 




2588 


9659 


2756 


9613 


2924 


9563 


3090 


9511 


3256 


9455 


2591 
2594 
2597 
2599 


9659 
9658 
9657 
9656 


2759 

2762 
2765 
2768 


9612 
9611 
9610 
9609 


2926 
2929 
2932 
2935 


9562 
9561 
9560 
9560 


3093 
3096 
3098 
3101 


9510 
9509 
9508 
9507 


3258 
3261 
3264 
3267 


9454 
9453 
9452 
9451 


2602 


9655 


2770 


9609 


2938 


9559 


3104 


9506 


3269 


9450 


2605 
2608 
2611 
2613 


9655 
9654 
9653 
9652 


2773 

2776 
2779 
2782 


9608 
9607 
9606 
9605 


2940 
2943 
2946 
2949 


9558 
9557 
9556 
9555 


3107 
3110 
3112 
3115 


9505 
9504 
9503 
9502 


3272 
3275 
3278 
3280 


9449 
9449 
9448 
9447 


2616 


9652 


2784 


9605 


2952 


9555 


3118 


9502 


3283 


9446 


2619 
2622 
2625 
2628 


9651 
9650 
9649 
9649 


2787 
2790 
2793 
2795 


9604 
9603 
9602 
9601 


2954 
2957 
2960 
2963 


9554 
9553 
9552 
9551 


3121 
3123 
3126 
3129 


9501 
9500 
9499 
9498 


3286 
3289 
3291 
3294 


9445 
9444 
9443 
9442 


2630 


9648 


2798 


9600 


2965 


9550 


3132 


9497 


3297 


9441 


2633 
2636 
2639 
2642 


9647 
9646 
9646 
9645 


2801 
2804 
2807 
2809 


9600 
9599 
9598 
9597 


2968 
2971 
2974 
2977 


9549 
9548 
9548 
9547 


3134 
3137 
3140 
3143 


9496 
9495 
9494 
9493 


3300 
3302 
3305 
3308 


9440 
9439 
9438 
9437 


2644 


9644 


2812 


9596 


2979 


9546 


3145 


9492 


3311 


9436 


2647 
2650 
2653 
2656 


9643 
9642 
9642 
9641 


2815 

2818 
2821 
2823 


9596 
9595 
9594 
9593 


2982 
2985 
2988 
2990 


9545 
9544 
9543 
9542 


3148 
3151 
3154 
3156 


9492 
9491 
9490 
9489 


3313 
3316 
3319 
3322 


9435 
9434 
9433 
9432 


2658 


9640 


2826 


9592 


2993 


9542 


3159 


9488 


3324 


9431 


2661 
2664 
2667 
2670 


9639 
9639 
9638 
9637 


2829 
2832 
2835 
2837 


9591 
9591 
9590 
9589 


2996 
2999 
3002 
3004 


9541 
9540 
9539 
9538 


3162 
3165 
3168 
3170 


9487 
9486 
9485 
9484 


3327 
3330 
3333 
3335 


9430 
9429 
9428 
9427 


2672 


9636 


2840 


9588 


3007 


9537 


3173 


9483 


3338 


9426 


2675 
2678 
2681 
2684 


9636 
9635 
9634 
9633 


2843 
2846 
2849 
2851 


9587 
9587 
9586 
9585 


3010 
3013 
3015 
3018 


9536 
9535 
9535 
9534 


3176 
3179 
3181 
3184 


9482 
9481 
9480 
9480 


3341 
3344 
3346 
3349 


9425 
9424 
9423 
9423 


2686 


9632 


2854 


9584 


3021 


9533 


3187 


9479 


3352 


9422 


2689 
2692 
2695 
2698 


9632 
9631 
9630 
9629 


2857 
2860 
2862 
2865 


9583 
9582 
9582 
9581 


3024 
3026 
3029 
3032 


9532 
9531 
9530 
9529 


3190 
3192 
3195 
3198 


9478 
9477 
9476 
9475 


3355 
3357 
3360 
3363 


9421 
9420 
9419 
9418 


2700 


9628 


2868 


9580 


3035 


9528 


3201 


9474 


3365 


9417 


2703 
2706 
2709 
2712 


9628 
9627 
9626 
9625 


2871 
2874 
2876 
2879 


9579 
9578 
9577 
9577 


3038 
3040 
3043 
3046 


9527 
9527 
9526 
9525 


3203 
3206 
3209 
3212 


9473 
9472 
9471 
9470 


3368 
3371 
3374 
3376 


9416 
9415 
9414 
9413 


2714 


9625 


2882 


9576 


3049 


9524 


3214 


9469 


3379 


9412 


2717 
2720 
2723 
2726 


9624 
9623 
9622 
9621 


2885 
2888 
2890 
2893 


9575 
9574 
9573 
9572 


3051 
3054 
3057 
3060 


9523 
9522 
9521 
9520 


3217 
3220 
3223 
3225 


9468 
9467 
9466 
9466 


3382 
3385 
3387 
3390 


9411 
9410 
9409 
9408 


2728 


9621 


2896 


9572 


3062 


9520 


3228 


9465 


3393 


9407 


2731 
2734 
2737 
2740 


9620 
9619 
9618 
9617 


2899 
2901 
2904 
2907 


9571 
9570 
9569 
9568 


3065 
3068 
3071 
3074 


9519 
9518 
9517 
9516 


3231 
3234 
3236 
3239 


9464 
9463 
9462 
9461 


3396 
3398 
3401 
3404 


9406 
9405 
9404 
9403 


2742 


9617 


2910 


9567 


3076 


9515 


3242 


9460 


3407 


9402 


2745 
2748 
2751 

2754 


9616 
9615 
9614 
9613 


2913 
2915 
2918 
2921 


9566 
9566 
9565 
9564 


3079 
3082 
3085 
3087 


9514 
9513 
9512 
9511 


3245 
3247 
3250 
3253 


9459 
9458 
9457 
9456 


3409 
3412 
3415 
3417 


9401 
9400 
9399 
9398 


2756 


9613 


2924 


9563 


3090 


9511 


3256 


9455 


3420 


9397 


COS 


sin 


COS 


sin 


COS 


sin 


COS 


sin 


COS 


sin 


t 


74° 


73° 


72° 


71 


o 


70° 


/ 



NATURAL SINES AND COSINES 



77 



1 


20° 


21° 


22° 


23° 


24° 


/ 1 




1 

2 
3 

4 

5 

6 
7 
8 
9 

10 

11 

12 
13 

14 

15 

16 
17 
18 
19 

20 

21 
22 
23 
24 

25 

26 

27 
28 
29 

30 

31 
32 
23 
34 

35 

36 
37 

38 
39 

40 

41 
42 
43 

44 

45 

46 
47 
48 
49 

50 

51 
52 
53 
54 

55 

56 
57 
58 
59 

60 


sin 


cos 


sin 


COS 


sin 


cos 


sin 


cos 


sin 


cos 


60 

59 
58 
57 
56 

55 
54 
53 
52 
51 

50 

49 
48 
47 
46 

45 

44 
43 
42 
41 

40 

39 
38 
37 
36 

35 

34 
33 
32 
31 

30 

29 
28 
27 
26 

25 

24 
23 
22 
21 

20 

19 
18 
17 

16 

15 

14 
13 
12 
11 

10 

9 
8 

7 
6 

5 

4 
3 

2 

1 




3420 


9397 


3584 


9336 


3746 


9272 


3907 


9205 


4067 


9135 


3423 
3426 
3428 
3431 


9396 
9395 
9394 
9393 


3586 
3589 
3592 
3595 


9335 
9334 
9333 
9332 


3749 
3751 
3754 
3757 


9271 
9270 
9269 
9267 


3910 
3913 
3915 
3918 


9204 
9203 
9202 
9200 


4070 
4073 
4075 
4078 


9134 
9133 
9132 
9131 


3434 


9392 


3597 


9331 


3760 


9266 


3921 


9199 


4081 


9130 


3437 
3439 
3442 
3445 


9391 
9390 
9389 
9388 


3600 
3603 
3605 
3608 


9330 
9328 
9327 
9326 


3762 
3765 
3768 
3770 


9265 
9264 
9263 
9262 


3923 
3926 
3929 
3931 


9198 
9197 
9196 
9195 


4083 
4086 
4089 
4091 


9128 
9127 
9126 
9125 


3448 


9387 


3611 


9325 


3773 


9261 


3934 


9194 


4094 


9124 


3450 
3453 
3456 
3458 


9386 
9385 
9384 
9383 


3614 
3616 
3619 
3622 


9324 
9323 
9322 
9321 


3776 
3778 
3781 
3784 


9260 
9259 
9258 
9257 


3937 
3939 
3942 
3945 


9192 
9191 
9190 
9189 


4097 
4099 
4102 
4105 


9122 
9121 
9120 
9119 


3461 


9382 


3624 


9320 


3786 


9255 


3947 


9188 


4107 


9118 


3464 
3467 
3469 
3472 


9381 
9380 
9379 
9378 


3627 
3630 
3633 
3635 


9319 
9318 
9317 
9316 


3789 
3792 
3795 
3797 


9254 
9253 
9252 
9251 


3950 
3953 
3955 
3958 


9187 
9186 
9184 
9183 


4110 
4112 
4115 
4118 


9116 
9115 
9114 
9113 


3475 


9377 


3638 


9315 


3800 


9250 


3961 


9182 


4120 


9112 


3478 
3480 
3483 
3486 


9376 
9375 
9374 
9373 


3641 
3643 
3646 
3649 


9314 
9313 
9312 
9311 


3803 
3805 
3808 
3811 


9249 
9248 
9247 
9245 


3963 
3966 
3969 
3971 


9181 
9180 
9179 
9178 


4123 
4126 
4128 
4131 


9110 
9109 
9108 
9107 


3488 


9372 


3651 


9309 


3813 


9244 


3974 


9176 


4134 


9106 


3491 
3494 
3497 
3499 


9371 
9370 
9369 
9368 


3654 
3657 
3660 
3662 


9308 
9307 
9306 
9305 


3816 
3819 
3821 
3824 


9243 
9242 
9241 
9240 


3977 
3979 
3982 
3985 


9175 
9174 
9173 
9172 


4136 
4139 
4142 

4144 


9104 
9103 
9102 
9101 


3502 


9367 


3665 


9304 


3827 


9239 


3987 


9171 


4147 


9100 


3505 
3508 
3510 
3513 


9366 
9365 
9364 
9363 


3668 
3670 
3673 
3676 


9303 
9302 
9301 
9300 


3830 
3832 
3835 
3838 


9238 
9237 
9235 
9234 


3990 
3993 
3995 
3998 


9169 
9168 
9167 
9166 


4150 
4152 
4155 
4158 


9098 
9097 
9096 
9095 


3516 


9362 


3679 


9299 


3840 


9233 


4001 


9165 


4160 


9094 


3518 
3521 
3524 
3527 


9361 
9360 
9359 
9358 


3681 
3684 
3687 
3689 


9298 
9297 
9296 
9295 


3843 
3846 
3848 
3851 


9232 
9231 
9230 
9229 


4003 
4006 
4009 
4011 


9164 
9162 
9161 
9160 


4163 
4165 
4168 
4171 


9092 
9091 
9090 
9088 


3529 


9356 


3692 


9293 


3854 


9228 


4014 


9159 


4173 


9088 


3532 
3535 
3537 
3540 


9355 
9354 
9353 
9352 


3695 
3697 
3700 
3703 


9292 
9291 
9290 
9289 


3856 
3859 
3862 
3864 


9227 
9225 
9224 
9223 


4017 
4019 
4022 
4025 


9158 
9157 
9155 
9154 


4176 
4179 
4181 
4184 


9086 
9085 
9084 
9083 


3543 


9351 


3706 


9288 


3867 


9222 


4027 


9153 


4187 


9081 


3546 
3548 
3551 
3554 


9350 
9349 
9348 
9347 


3708 
3711 
3714 
3716 


9287 
9286 
9285 
9284 


3870 

3872 
3875 
3878 


9221 
9220 
9219 
9218 


4030 
4033 
4035 
4038 


9152 
9151 
9150 
9148 


4189 
4192 
4195 
4197 


9080 
9079 
9078 
9077 


3557 


9346 


3719 


9283 


3881 


9216 


4041 


9147 


4200 


9075 


3559 
3562 
3565 
3567 


9345 
9344 
9343 
9342 


3722 
3724 
3727 
3730 


9282 
9281 
9279 
9278 


3883 
3886 
3889 
3891 


9215 
9214 
9213 
9212 


4043 
4046 
4049 
4051 


9146 
9145 
9144 
9143 


4202 
4205 
4208 
4210 


9074 
9073 
9072 
9070 


3570 


9341 


3733 


9277 


3894 


9211 


4054 


9141 


4213 


9069 


3573 
3576 
3578 
3581 


9340 
9339 
9338 
9337 


3735 
3738 
3741 
3743 


9276 
9275 
9274 
9273 


3897 
3899 
3902 
3905 


9210 
9208 
9207 
9206 


4057 
4059 
4062 
4065 


9140 
9139 
9138 
9137 


4216 
4218 
4221 
4224 


9068 
9067 
9066 
9064 


3584 


9336 


3746 


9272 


3907 


9205 


4067 


9135 


4226 


9063 


COS 


sin 


COS 


sin 


cos 


sin 


COS 


sin 


COS 


sin 


/ 


69° 


68° 


67° 


66° 


65° 


/ 



7 8 



NATURAL SINES AND COSINES 



/ 


25° 


26° 


27° 


28° 


29° 


i 




1 

2 
3 
4 

5 

6 

7 
8 
9 

10 

11 

12 
13 
14 

15 

16 
17 
18 
19 

20 

21 
22 
23 
24 

25 

26 

27 
28 
29 

30 

31 
32 
33 
34 

35 

36 
37 
38 
39 

40 

41. 

42 

43 

44 

45 

46 
47 
48 
49 

50 

51 
52 
53 
54 

55 

56 
57 
58 
59 

60 


sin 


cos 


sin 


COS 


sin 


cos 


sin 


cos 


sin 


cos 


60 

59 
58 
57 
56 

55 

54 
53 
52 
51 
50 

49 
48 
47 
46 

45 

44 
43 
42 
41 

40 

39 
38 
37 
36 

35 

34 
33 
32 
31 
30 
29 
28 
27 
26 

25 

24 
23 
22 
21 

20 

19 
18 
17 
16 
15 
14 
13 
12 
11 

10 

9 

8 
7 
6 

5 

4 
3 
2 
1 




4226 


9063 


4384 


8988 


4540 


8910 


4695 


8829 


4848 


8746 • 


4229 
4231 
4234 
4237 


9062 
9061 
9059 
9058 


4386 
4389 
4392 
4394 


8987 
8985 
8984 
8983 


4542 
4545 
4548 
4550 


8909 
8907 
8906 
8905 


4697 
4700 
4702 
4705 


8828 
8827 
8825 
8824 


4851 
4853 
4856 
4858 


8745 
8743 
8742 
8741 


4239 


9057 


4397 


8982 


4553 


8903 


4708 


8823 


4861 


8739 


4242 
4245 
4247 
4250 


9056 
9054 
9053 
9052 


4399 
4402 
4405 
4407 


8980 
8979 
8978 
8976 


4555 
4558 
4561 
4563 


8902 
8901 
8899 
8898 


4710 
4713 
4715 
4718 


8821 
8820 
8819 
8817 


4863 
4866 
4868 
4871 


8738 
8736 
8735 
8733 


4253 


9051 


4410 


8975 


4566 


8897 


4720 


8816 


4874 


8732 


4255 
4258 
4260 
4263 


9050 
9048 
9047 
9046 


4412 
4415 
4418 
4420 


8974 
8973 
8971 
8970 


4568 
4571 
4574 
4576 


8895 
8894 
8893 
8892 


4723 
4726 
4728 
4731 


8814 
8813 
8812 
8810 


4876 
4879 
4881 
4884 


8731 
8729 
8728 
8726 


4266 


9045 


4423 


8969 


4579 


8890 


4733 


8809 


4886 


8725 


4268 
4271 
4274 
4276 


9043 
9042 
9041 
9040 


4425 
4428 
4431 
4433 


8967 
8966 
8965 
8964 


4581 
4584 
4586 
4589 


8889 
8888 
8886 
8885 


4736 
4738 
4741 
4743 


8808 
8806 
8805 
8803 


4889 
4891 
4894 
4896 


8724 
8722 
8721 
8719 


4279 


9038 


4436 


8962 


4592 


8884 


4746 


8802 


4899 


8718 


4281 
4284 
4287 
4289 


9037 
9036 
9035 
9033 


4439 
4441 
4444 
4446 


8961 
8960 
8958 
8957 


4594 
4597 
4599 
4602 


8882 
8881 
8879 
8878 


4749 
4751 
4754 
4756 


8801 
8799 
8798 
8796 


4901 
4904 
4907 
4909 


8716 
8715 
8714 
8712 


4292 


9032 


4449 


8956 


4605 


8877 


4759 


8795 


4912 


8711 


4295 
4297 
4300 
4302 


9031 
9030 
9028 
9027 


4452 
4454 
4457 
4459 


8955 
8953 
8952 
8951 


4607 
4610 
4612 
4615 


8875 
8874 
8873 
8871 


4761 
4764 
4766 
4769 


8794 
8792 
8791 
8790 


4914 
4917 
4919 
4922 


8709 
8708 
8706 
8705 


4305 


9026 


4462 


8949 


4617 


8870 


4772 


8788 


4924 


8704 


4308 
4310 
4313 
4316 


9025 
9023 
9022 
9021 


4465 
4467 
4470 
4472 


8948 
8947 
8945 
8944 


4620 
4623 
4625 
4628 


8869 
8867 
8866 
8865 


4774 
4777 
4779 
4782 


8787 
8785 
8784 
8783 


4927 
4929 
4932 
4934 


8702 
8701 
8699 
8698 


4318 


9020 


4475 


8943 


4630 


8863 


4784 


8781 


4937 


8696 


4321 
4323 
4326 
4329 


9018 
9017 
9016 
9015 


4478 
4480 
4483 
4485 


8942 
8940 
8939 
8938 


4633 
4636 
4638 
4641 


8862 
8861 
8859 
8858 


4787 
4789 
4792 
4795 


8780 
8778 
8777 
8776 


4939 
4942 
4944 
4947 


8695 
8694 
8692 
8691 


4331 


9013 


4488 


8936 


4643 


8857 


4797 


8774 


4950 


8689 


4334 
4337 
4339 
4342 


9012 
9011 
9010 
9008 


4491 
4493 
4496 
4498 


8935 
8934 
8932 
8931 


4646 
4648 
4651 
4654 


8855 
8854 
8853 
8851 


4800 
4802 
4805 
4807 


8773 
8771 
8770 
8769 


4952 
4955 
4957 
4960 


8688 
8686 
8685 
8683 


4344 


9007 


4501 


8930 


4656 


8850 


4810 


8767 


4962 


8682 


4347 
4350 
4352 
4355 


9006 
9004 
9003 
9002 


4504 
4506 
4509 
4511 


8928 
8927 
8926 
8925 


4659 
4661 
4664 
4666 


8849 
8847 
8846 
8844 


4812 
4815 
4818 
4820 


8766 
8764 
8763 
8762 


4965 
4967 
4970 
4972 


8681 
8679 
8678 
8676 


4358 


9001 


4514 


8923 


4669 


8843 


4823 


8760 


4975 


8675 


4360 
4363 
4365 
4368 


8999 
8998 
8997 
8996 


4517 
4519 
4522 
4524 


8922 
8921 
8919 
8918 


4672 
4674 
4677 
4679 


8842 
8840 
8839 
8838 


4825 
4828 
4830 
4833 


8759 
8757 
8756 

8755 


4977 
4980 
4982 
4985 


8673 
8672 
8670 
8669 


4371 


8994 


4527 


8917 


4682 


8836 


4835 


8753 


4987 


8668 


4373 
4376 
4378 
4381' 


8993 
8992 
8990 
8989 


4530 
4532 
4535 
4537 


8915 
8914 
8913 
8911 


4684 
4687 
4690 
4692 


8835 
8834 
8832 
8831 


4838 
4840 
4843 
4846 


8752 
8750 
8749 
8748 


4990 
4992 
4995 
4997 


8666 
8665 
8663 
8662 


4384 


8988 


4540 


8910 


4695 


8829 


4848 


8746 


5000 


8660 


cos 


sin 


COS 


sin 


COS 


sin 


COS 


sin 


COS 


sin 


/ 


64° 


63° 


62° 


61° 


60° 


/ 







-— 



__ 



NATURAL SINES AND COSINES 



79 



/ 


30° 


31° 


32° 


33° 


34° 


t 




1 

2 
3 

4 

5 

6 

7 
8 
9 

10 

11 

12 
13 
14 

15 

16 
17 
18 
19 

20 

21 
22 
23 

24 

25 

26 
27 
28 
29 

30 

31 
32 
33 
34 

35 

36 
37 
38 
39 


sin 


cos 


sin 


COS 


sin 


cos 


sin 


cos 


sin 


cos 


60 

59 
58 
57 
56 
55 
54 
58 
52 
51 
50 
49 
48 
47 
46 

45 

44 
43 
42 
41 

40 

39 
38 
37 
36 
35 
34 
33 
32 
31 

30 

29 
28 
27 
26 

25 

24 
23 
22 
21 

20 

19 
18 
17 
16 

15 

14 
13 
12 
11 

10 

9 
8 
7 
6 

5 

4 
3 
2 
1 




5000 


8660 


5150 


8572 


5299 


8480 


5446 


8387 


5592 


8290 


5003 
5005 
5008 
5010 


8659 
8657 
8656 
8654 


5153 
5155 
5158 
5160 


8570 
8569 
8567 
8566 


5302 
5304 
5307 
5309 


8479 
8477 
8476 
8474 


5449 
5451 

5454 
5456 


8385 
8384 
8382 
8380 


5594 
5597 
5599 
5602 


8289 
8287 
8285 
8284 


5013 


8653 


5163 


8564 


5312 


8473 


5459 


8379 


5604 


8282 


5015 
5018 
5020 
5023 


8652 • 
8650 
8649 
8647 


5165 
5168 
5170 
5173 


8563 
8561 
8560 
8558 


5314 
5316 
5319 
5321 


8471 
8470 
8468 
8467 


5461 
5463 
5466 
5468 


8377 
8376 
8374 
8372 


5606 
5609 
5611 
5614 


8281 
8279 
8277 
8276 


5025 


8646 


5175 


8557 


5324 


8465 


5471 


8371 


5616 


8274 


5028 
5030 
5033 
5035 


8644 
8643 
8641 
8640 


5178 
5180 
5183 
5185 


8555 
8554 
8552 
8551 


5326 
5329 
5331 
5334 


8463 
8462 
8460 
8459 


5473 
5476 
5478 
5480 


8369 
8368 
8366 
8364 


5618 
5621 
5623 
5626 


8272 
8271 
8269 
8268 


5038 


8638 


5188 


8549 


5336 


8457 


5483 


8363 


5628 


8266 


5040 
5043 
5045 
5048 


8637 
8635 
8634 
8632 


5190 
5193 
5195 
5198 


8548 
8546 
8545 
8543 


5339 

5341 

I 5344 

5346 


8456 
8454 
8453 
8451 


5485 
5488 
5490 
5493 


8361 
8360 
8358 
8356 


5630 
5633 
5635 
5638 


8264 
8263 
8261 
8259 


5050 


8631 


5200 


8542 


5348 


8450 


5495 


8355 


5640 


8258 


5053 
5055 
5058 
5060 


8630 
8628 
8627 
8625 


5203 
5205 
5208 
5210 


8540 
8539 
8537 
8536 


5351 
5353 
5356 
5358 


8448 
8446 
8445 
8443 


5498 
5500 
5502 
5505 


8353 
8352 
8350 
8348 


5642 
5645 
5647 
5650 


8256 
8254 
8253 
8251 


5063 


8624 


5213 


8534 


5361 


8442 


5507 


8347 


5652 


8249 


5065 
5068 
5070 
5073 


8622 
8621 
8619 

8618 


5215 
5218 
5220 
5223' 


8532 
8531 
8529 
8528 


5363 
5366 
5368 
5371 


8440 
8439 
8437 
8435 


5510 
5512 
5515 
5517 


8345 
8344 
8342 
8340 


5654 
5657 
5659 
5662 


8248 
8246 
8245 
8243 


5075 


8616 


5225 


8526 


5373 


8434 


5519 


8339 


5664 


8241 


5078 
5080 
5083 
5085 


8615 
8613 
8612 
8610 


5227 
5230 
5232 
5235 


8525 
8523 
8522 
8520 


5375 
5378 
5380 
5383 


8432 
8431 
8429 
8428 


5522 
5524 
5527 
5529 


8337 
8336 
8334 
8332 


5666 
5669 
5671 
5674 


8240 
8238 
8236 
8235 


5088 


8609 


5237 


8519 


5385 


8426 


5531 


8331 


5676 


8233 


5090 
5093 
5095 
5098 


8607 
8606 
8604 
8603 


5240 
5242 
5245 
5247 


8517 
8516 
8514 
8513 


5388 
5390 
5393 
5395 


8425 
8423 
8421 
8420 


5534 
5536 
5539 
5541 


8329 
8328 
8326 
8324 


5678 
5681 
5683 
5686 


8231 
8230 

8228 
8226 


40 


5100 


8601 


5250 


8511 


5398 


8418 


5544 


8323 


5688 


8225 


41 

42 
43 
44 

45 

46 
47 
48 
49 

50 

51 
52 
53 
54 

55 

56 
57 
58 
59 

60 


5103 
5105 
5108 
5110 


8600 
8599 
8597 
8596 


5252 
5255 
5257 
5260 


8510 
8508 
8507 
8505 


5400 
5402 
5405 
5407 


8417 
8415 
8414 
8412 


5546 
5548 
5551 
5553 


8321 
8320 
8318 
8316 


5690 
5693 
5695 
5698 


8223 
8221 
8220 
8218 


5113 


8594 


5262 


8504 


5410 


8410 


5556 


8315 


5700 


8216 


5115 
5118 
5120 
5123 


8593 
8591 
8590 
8588 


5265 
5267 
5270 
5272 


8502 
8500 
8499 
8497 


5412 
5415 
5417 
5420 


8409 
8407 
8406 
8404 


5558 
5561 
5563 
5565 


8313 
8311 
8310 
8308 


5702 
5705 
5707 
5710 


8215 
8213 
8211 
8210 


5125 


8587 


5275 


8496 


5422 


8403 


5568 


8307 


5712 


8208 


5128 
5130 
5133 
5135 


8585 
8584 
8582 
8581 


5277 
5279 
5282 
5284 


8494 
8493 
8491 
8490 


5424 
5427 
5429 
5432 


8401 
8399 
8398 
8396 


5570 
5573 
5575 
5577 


8305 
8303 
8302 
8300 


5714 

5717 
5719 
5721 


8207 
8205 
8203 
8202 


5138 


8579 


5287 


8488 


5434 


8395 


5580 


8299 


5724 


8200 


5140 
5143 
5145 
5148 


8578 
8576 
8575 
8573 


5289 
5292 
5294 
5297 


8487 
8485 
8484 
8482 


5437 
5439 
5442 
5444 


8393 
8391 
8390 
8388 


5582 
5585 
5587 
5590 


8297 
8295 
8294 
8292 


5726 
5729 
5731 
5733 


8198 
8197 
8195 
8193 


5150 


8572 


5299 


8480 


5446 


8387 


5592 


8290 


5736 


8192 


COS 


sin 


COS 


sin 


cos 


sin 


COS 


sin 


COS 


sin 


/ 


59° 


58° 


57° 


56° 


55° 


f 



8o 



NATURAL SINES AND COSINES 



/ 


35° 


36° 


37° 


38° 


39° 


/ 




1 

2 
3 

4 

5 

6 

7' 

8 

9 

10 

11 
12 
13 

14 

15 

16 
17 
18 
19 

20 

21 
22 
23 

24 

25 

26 
27 
28 
29 

30 

31 
32 
33 

34 

35 

36 
37 
38 
39 

40 

41 
42 
43 

44 

45 

46 
47 
48 
49 

50 

51 
52 
53 
54 

55 

56 
57 
58 
59 

60 


sin 


cos 


sin 


cos 


sin 


cos 


sin 


cos 


sin 


cos 


60 

59 
58 
57 
56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 
43 
42 
41 

40 

39 
38 
37 
36 

35 

34 
33 
32 
31 

30 
29 
28 
27 
26 

25 
24 
23 
22 
21 

20 

19 
18 
17 
16 

15 

14 
13 
12 
11 

10 

9 

8 
7 
6 

5 

4 
3 
2 
1 




5736 


8192 


5878 


8090 


6018 


7986 


6157 


7880 


6293 


7771 


5738 
5741 
5743 
5745 


8190 
8188 
8187 
8185 


5880 
5883 

5885 
5887 


8088 
8087 
8085 
8083 


6020 
6023 
6025 
6027 


7985 
7983 
7981 
7979 


6159 
6161 
6163 
6166 


7878 

7877 
7875 
7873 


6295 
6298 
6300 
6302 


7770 
7768 
7766 
7764 


5748 


8183 


5890 


8082 


6030 


7978 


6168 


7871 


6305 


7762 


5750 
5752 
5755 
5757 


8181 
8180 
8178 
8176 


5892 
5894 
5897 
5899 


8080 
8078 
8076 
8075 


6032 
6034 
6037 
6039 


7976 
7974 
7972 
7971 


6170 
6173 
6175 
6177 


7869 
7868 
7866 
7864 


6307 
6309 
6311 
6314 


7760 
7759 

7757 
7755 


5760 


8175 


5901 


8073 


6041 


7969 


6180 


7862 


6316 


7753 


5762 
5764 
5767 
5769 


8173 
8171 
8170 
8168 


5904 
5906 
5908 
5911 


8071 
8070 
8068 
8066 


6044 
6046 
6048 
6051 


7967 
7965 
7964 
7962 


6182 
6184 
6186 
6189 


7860 
7859 

7857 
7855 


6318 
6320 
6323 
6325 


7751 

7749 
7748 
7746 


5771 


8166 


5913 


8064 


6053 


7960 


6191 


7853 


6327 


7744 


5774 
5776 
5779 
5781 


8165 
8163 
8161 
8160 


5915 
5918 
5920 
5922 


8063 
8061 
8059 
8058 


6055 
6058 
6060 
6062 


7958 
7956 
7955 
7953 


6193 
6196 
6198 
6200 


7851 
7850 
7848 
7846 


6329 
6332 
6334 
6336 


7742 
7740 
7738 
7737 


5783 


8158 


5925 


8056 


6065 


7951 


6202 


7844 


6338 


7735 


5786 
5788 
5790 
5793 


8156 
8155 
8153 
8151 


5927 
5930 
5932 
5934 


8054 
8052 
8051 
8049 


6067 
6069 
6071 
6074 


7950 
7948 
7946 
7944 


6205 
6207 
6209 
6211 


7842 
7841 
7839 
7837 


6341 
6343 
6345 
6347 


7733 
7731 

7729 

7727 


5795 


8150 


5937 


8047 


6076 


7942 


6214 


7835 


6350 


7725 


5798 
5800 
5802 
5805 


8148 
8146 
8145 
8143 


5939 
5941 
5944 
5946 


8045 
8044 
8042 
8040 


6078 
6081 
6083 
6085 


7941 
7939 
7937 
7935 


6216 
6218 
6221 
6223 


7833 
7832 
7830 

7828 


6352 
6354 
6356 
6359 


7724 
7722 
7720 
7718 


5807 


8141 


5948 


8039 


6088 


7934 


6225 


7826 


6361 


7716 


5809 
5812 
5814 
5816 


8139 
8138 
8136 
8134 


5951 
5953 
5955 
5958 


8037 
8035 
8033 
8032 


6090 
6092 
6095 
6097 


7932 
7930 
7928 
7926 


6227 
6230 
6232 
6234 


7824 
7822 
7821 
7819 


6363 
6365 
6368 
6370 


7714 
7713 
7711 
7709 


5819 


8133 


5960 


8030 


6099 


7925 


6237 


7817- 


6372 


7707 


5821 
5824 
5826 
5828 


8131 
8129 
8128 
8126 


5962 
5965 
5967 
5969 


8028 
8026 
8025 
8023 


6101 
6104 
6106 
6108 


7923 
7921 
7919 
7918 


6239 
6241 
6243 
6246 


7815 
7813 
7812 
7810 


6374 
6376 
6379 
6381 


7705 
7703 
7701 
7700 


5831 


8124 


5972 


8021 


6111 


7916 


6248 


7808 


6383 


7698 


5833 
5835 
5838 
5840 


8123 
8121 
8119 
8117 


5974 
5976 
5979 
5981 


8020 
8018 
8016 
8014 


6113 
6115 
6118 
6120 


7914 
7912 
7910 
7909 


6250 
6252 
6255 
6257 


7806 
7804 
7802 
7801 


6385 
6388 
6390 
6392 


7696 
7694 
7692 
7690 


5842 


8116 


5983 


8013 


6122 


7907 


6259 


7799 


6394 


7688 


5845 
5847 
5850 
5852 


8114 
8112 
8111 
8109 


5986 
5988 
5990 
5993 


8011 
8009 
8007 
8006 


6124 
6127 
6129 
6131 


7905 
7903 
7902 
7900 


6262 
6264 
6266 
6268 


7797 
7795 
7793 
7792 


6397 
6399 
6401 
6403 


7687 
7685 
7683 
7681 


5854 


8107 


5995 


8004 


6134 


7898 


6271 


7790 


6406 


7679 


5857 
5859 
5861 
5864 


8106 
8104 
8102 
8100 


5997 
6000 
6002 
6004 


8002 
8000 
7999 
7997 


6136 
6138 
6141 
6143 


7896 
7894 
7893 
7891 


6273 
6275 
6277 
6280 


7788 
7786 
7784 
7782 


6408 
6410 
6412 
6414 


7677 
7675 
7674 
7672 


5866 


8099 


6007 


7995 


6145 


7889 


6282 


7781 


6417 


7670 


5868 
5871 
5873 
5875 


8097 
8095 
8094 
8092 


6009 
6011 
6014 
6016 


7993 
7992 
7990 
7988 


6147 
6150 
6152 
6154 


7887 
7885 
7884 
7882 


6284 
6286 
6289 
6291 


7779 
7777 
7775 
7773 


6419 
6421 
6423 
6426 


7668 
7666 
7664 
7662 


5878 


8090 


6018 


7986 


6157 


7880 


6293 


7771 


6428 


7660 


COS 


sin 


cos 


sin 


cos 


sin 


COS 


sin 


COS 


sin 


/ 


54° 


53° 


52° 


51° 


50° 


i 



NATURAL SINES AND COSINES 



8l 



1 


40° 


41° 


42° 


43° 


44° 


/ 




1 

2 
3 
4 

5 

6 

7 
8 
9 

10 

11 
12 
13 
14 

15 

16 

17 
18 
19 

20 

21 
22 
23 
24 

25 

26 

27 
28 
29 

30 

31 
32 
33 
34 

35 

36 
37 
38 
39 

40 

41 
42 
43 
44 

45 

46 

47 
48 
49 

50 

51 
52 
53 
54 
55 
56 
57 
58 
59 

60 


sin 


cos 


sin 


cos 


sin 


cos 


sin 


cos 


sin 


cos 


60 

59 
58 
57 
56 

55 

54 
53 
52 
51 

50 

49 
48 
47 
46 

45 

44 
43 
42 
41 

40 

39 
38 
37 
36 

35 

34 
33 
32 
31 

30 

29 
28 
27 
26 

25 

24 
23 
22 
21 

20 

19 
18 
17 
16 

15 

14 
13 
12 
11 

10 

9 
8 
7 
6 

5 

4 
3 
2 
1 




6428 


7660 


6561 


7547 


6691 


7431 


6820 


7314 


6947 


7193 


6430 
6432 
6435 
6437 


7659 
7657 
7655 
7653 


6563 
6565 
6567 
6569 


7545 
7543 
7541 
7539 


6693 
6696 
6698 
6700 


7430 
7428 
7426 
7424 


6822 
6824 
6826 
6828 


7312 
7310 
7308 
7306 


6949 
6951 
6953 
6955 


7191 
7189 
7187 
7185 


6439 


7651 


6572 


7538 


6702 


7422 


6831 


7304 


6957 


7183 


6441 
6443 
6446 
6448 


7649 
7647 
7645 
7644 


6574 
6576 
6578 
6580 


7536 
7534 
7532 
7530 


6704 
6706 
6709 
6711 


7420 
7418 
7416 
7414 


6833 
6835 
6837 
6839 


7302 
7300 
7298 
7296 


6959 
6961 
6963 
6965 


7181 
7179 
7177 
7175 


6450 


7642 


6583 


7528 


6713 


7412 


6841 


7294 


6967 


7173 


6452 
6455 
6457 
6459 


7640 
7638 
7636 
7634 


6585 
6587 
6589 
6591 


7526 
7524 
7522 
7520 


6715 
6717 
6719 
6722 


7410 
7408 
7406 
7404 


6843 
6845 
6848 
6850 


7292 
7290 
7288 
7286 


6970 
6972 
6974 
6976 


7171 
7169 
7167 
7165 


6461 


7632 


6593 


7518 


6724 


7402 


6852 


7284 


6978 


7163 


6463 
6466 
6468 
6470 


7630 
7629 
7627 
7625 


6596 
6598 
6600 
6602 


7516 
7515 
7513 
7511 


6726 
6728 
6730 
6732 


7400 
7398 
7396 
7394 


6854 
6856 
6858 
6860 


7282 
7280 
7278 
7276 


6980 
6982 
6984 
6986 


7161 
7159 
7157 
7155 


6472 


7623 


6604 


7509 


6734 


7392 


6862 


7274 


6688 


7153 


6475 
6477 
6479 
6481 


7621 
7619 
7617 
7615 


6607 
6609 
6611 
6613 


7507 
7505 
7503 
7501 


6737 
6739 
6741 
6743 


7390 
7388 
7387 
7385 


6865 
6867 
6869 
6871 


7272 
7270 
7268 
7266 


6990 
6992 
6995 
6997 


7151 
7149 
7147 
7145 


6483 


7613 


6615 


7499 


6745 


7383 


6873 


7264 


6999 


7143 


6486 
6488 
6490 
6492 


7612 
7610 
7608 
7606 


6617 
6620 
6622 
6624 


7497 
7495 
7493 
7491 


6747 
6749 
6752 
6754 


7381 
7379 

7377 
7375 


6875 
6877 
6879 
6881 


7262 
7260 
7258 
7256 


7001 
7003 
7005 
7007 


7141 
7139 
7137 
7135 


6494 


7604 


6626 


7490 


6756 


7373 


6884 


7254 


7009 


7133 


6497 
6499 
6501 
6503 


7602 
7600 
7598 
7596 


6628 
6631 
6633 
6635 


7488 
7486 
7484 
7482 


6758 
6760 
6762 
6764 


7371 
7369 
7367 
7365 


6886 
6888 
6890 
6892 


7252 
7250 
7248 
7246 


7011 
7013 
7015 
7017 


7130 
7128 
7126 
7124 


6506 


7595 


6637 


7480 


6767 


7363 


6894 


7244 


7019 


7122 


6508 
6510 
6512 
6514 


7593 
7591 
7589 

7587 


6639 
6641 
6644 
6646 


7478 
7476 
7474 
7472 


6769 
6771 
6773 
6775 


7361 
7359 
7357 
7355 


6896 
6898 
6900 
6903 


7242 
7240 
7238 
7236 


7022 
7024 
7026 
7028 


7120 
7118 
7116 
7114 


6517 


7585 


6648 


7470 


6777 


7353 


6905 


7234 


7030 


7112 


6519 
6521 
6523 
6525 


7583 
7581 
7579 
7578 


6650 
6652 
6654 
6657 


7468 
7466 
7464 
7463 


6779 
6782 
6784 
6786 


7351 
7349 
7347 
7345 


6907 
6909 
6911 
6913 


7232 
7230 
7228 
7226 


7032 
7034 
7036 
7038 


7110 
7108 
7106 
7104 


6528 


7576 


6659 


7461 


6788 


7343 


6915 


7224 


7040 


7102 


6530 
6532 
6534 
6536 


7574 
7572 
7570 
7568 


6661 
6663 
6665 
6667 


7459 
7457 
7455 
7453 


6790 
6792 
6794 
6797 


7341 
7339 
7337 
7335 


6917 
6919 
6921 
6924 


7222 
7220 
7218 
7216 


7042 
7044 
7046 
7048 


7100 
7098 
7096 
7094 


6539 


7566 


6670 


7451 


6799 


7333 


6926 


7214 


7050 


7092 


6541 
6543 
6545 
6547 


7564 
7562 
7560 
7559 


6672 
6674 
6676 
6678 


7449 
7447 
7445 
7443 


6801 
6803 
6805 
6807 


7331 
7329 
7327 
7325 


6928 
6930 
6932 
6934 


7212 
7210 
7208 
7206 


7053 
7055 
7057 
7059 


7090 
7088 
7085 
7083 


6550 


7557 


6680 


7441 


6809 


7323 


6936 


7203 


7061 


7081 


6552 
6554 
6556 
6558 


7555 
7553 
7551 
7549 


6683 
6685 
6687 
6689 


7439 
7437 
7435 
7433 


6811 
6814 
6816 
6818 


7321 
7319 
7318 
7316 


6938 
6940 
6942 
6944 


7201 
7199 
7197 
7195 


7063 
7065 
7067 
7069 


7079 
7077 
7075 
7073 


6561 


7547 


6691 


7431 


6820 


7314 


6947 


7193 


7071 


7071 


COS 


sin 


COS 


sin 


cos 


sin 


cos 


sin 


cos 


sin 


i 


49° 


48° 


47° 


46° 


45° 


i 



82 



NATURAL TANGENTS AND COTANGENTS 





0° 


1° 


3° 


3° 


4° 


/ 




1 

2 
3 
4 

5 

6 
7 
8 
9 

10 

11 
12 
13 
14 

15 

16 

17 
18 
19 

30 

21 
22 
23 

24 

35 

26 
27 
28 
29 

30 

31 
32 
33 
34 

35 

36 
37 
38 
39 

40 

41 

42 
43 

44 

45 

46 
47 
48 
49 

50 

51 
52 
53 
54 

55 
56 
57 
58 
59 

60 


tan cot 


tan cot 


tan cot 


tan cot 


tan 


cot 


60 

59 
58 
57 
56 

55 

54 
53 
52 
51 
50 
49 
48 
47 
46 

45 

44 
43 
42 
41 

40 

39 
38 
37 
36 

35 

34 
33 
32 
31 
30 
29 
28 
27 
26 

35 

24 
23 
22 
21 

30 

19 
18 
17 
16 

15 

14 
13 
12 
11 

10 

9 

8 
7 
6 

5 

4 
3 
2 
1 




0000 Infinite 


0175 57.2900 


0349 28.6363 


0524 19.0811 


0699 


14.3007 


0003 3437.75 
0006 1718.87 
0009 1145.92 
0012 859.436 


0177 56.3506 
0180 55.4415 
0183 54.5613 
0186 53.7086 


0352 28.3994 
0355 28.1664 
0358 27.9372 
0361 27.7117 


0527 18.9755 
0530 18.8711 
0533 18.7678 
0536 18.6656 


0702 
0705 
0708 
0711 


14.2411 
14.1821 
14.1235 
14.0655 


0015 687.549 

0017 572.957 
0020 491.106 
0023 429.718 
0026 381.971 


0189 52.8821 


0364 27.4899 


0539 18.5645 


0714 


14.0079 


0192 52.0807 
0195 51.3032 
0198 50.5485 
0201 49.8157 


0367 27.2715 
0370 27.0566 
0373 26.8450 
0375 26.6367 


0542 18.4645 
0544 18.3655 
0547 18.2677 
0650 18.1708 


0717 
0720 
0723 
0726 


13.9507 
13.8940 
13.8378 
13.7821 


0029 343 . 774 


0204 49 . 1039 


0378 26.4316 


0553 18.0750 


0729 


13.7267 


0032 312.521 
0035 286.478 
0038 264.441 
0041 245.552 


0207 48.4121 
0209 47.7395 
0212 47.0853 
0215 46.4489 


0381 26.2296 
0384 26.0307 
0387 25.8348 
0390 25.6418 


0556 17.9802 
0559 17.8863 
0562 17.7934 
0565 17.7015 


0731 
0734 
0737 
0740 


13.6719 
13.6174 
13.5634 
13.5098 


0044 229.182 


0218 45.8294 


0393 25.4517 


0568 17.6106 


0743 


13.4566 


0047 214.858 
0049 202.219 
0052 190.984 
0055 180.932 


0221 45.2261 
0224 44.6386 
0227 44.0661 
0230 43.5081 


0396 25.2644 
0399 25.0798 
0402 24.8978 
0405 24.7185 


0571 17.5205 
0574 17.4314 
0577 17.3432 
0580 17.2558 


0746 
0749 
0752 
0755 


13.4039 
13.3515 
13.2996 
13.2480 


0058 171.885 


0233 42.9641 


0407 24.5418 


0582 17.1693 


0758 


13.1969 


0061 163.700 
0064 156.259 
0067 149.465 
0070 143.237 


0236 42.4335 
0239 41.9158 
0241 41.4106 
0244 40.9174 


0410 24.3675 
0413 24.1957 
0416 24.0263 
0419 23.8593 


0585 17.0837 
0588 16.9990 
0591 16.9150 
0594 16.8319 


0761 
0764 
0767 
0769 


13.1461 
13.0958 
13.0458 
12.9962 


0073 137.507 


0247 40.4358 


0422 23.6945 


0597 16.7496 


0772 


12.9469 


0076 132.219 
0079 127.321 
0081 122.774 
0084 118.540 


0250 39.9655 
0253 39.5059 
0256 39.0568 
0259 38.6177 


0425 23.5321 
0428 23.3718 
0431 23.2137 
0434 23.0577 


0600 16.6681 
0603 16.5874 
0606 16.5075 
0609 16.4283 


0775 
0778 
0781 
0784 


12.8981 
12.8496 
12.8014 
12.7536 


0087 114.589 


0262 38.1885 


0437 22.9038 


0612 16.3499 


0787 


12.7062 


0090 110.892 
0093 107.426 
0096 104.171 
0099 101 . 107 


0265 37.7686 
0268 37.3579 
0271 36.9560 
0274 36.5627 


0440 22.7519 
0442 22.6020 
0445 22.4541 
0448 22.3081 


0615 16.2722 
0617 16.1952 
0620 16.1190 
0623 16.0435 


0790 
0793 
0796 
0799 


12.6591 
12.6124 
12.5660 
12.5199 


0102 98.2179 


0276 36.1776 


0451 22.1640 


0626 15.9687 


0802 


12.4742 


0105 95.4895 
0108 92.9085 
0111 90.4633 
0113 88.1436 


0279 35.8006 
0282 35.4313 
0285 35.0695 
0288 34.7151 


0454 22.0217 
0457 21.8813 
0460 21.7426 
0463 21.6056 


0629 15.8945 
0632 15.8211 
0635 15.7483 
0638 15.6762 


0805 
0808 
0810 
0813 


12.4288 
12.3838 
12.3390 
12.2946 


0116 85.9398 


0291 34.3678 


0466 21.4704 


0641 15.6048 


0816 


12.2505 


0119 83.8435 
0122 81.8470 
0125 79.9434 
0128 78.1263 


0294 34.0273 
0297 33.6935 
0300 33.3662 
0303 33.0452 


0469 21.3369 
0472 21.2049 
0475 21.0747 
0477 20.9460 


0644 15.5340 
0647 15.4638 
0650 15.3943 
0653 15.3254 


0819 
0822 
0825 
0828 


12.2067 
12.1632 
12.1201 
12.0772 


0131 76.3900 


0306 32.7303 


0480 20.8188 


0655 15.2571 


0831 


12 . 0346 


0134 74.7292 
0137 73.1390 
0140 71.6151 
0143 70.1533 


0308 32.4213 
0311 32.1181 
0314 31.8205 
0317 31.5284 


0483 20.6932 
0486 20.5691 
0489 20.4465 
0492 20.3253 


0658 15.1893 
0661 15.1222 
0664 15.0557 
0667 14.9898 


0834 
0837 
0840 
0843 


11.9923 
11.9504 
11.9087 
11.8673 


0146 68.7501 


0320 31.2416 


0495 20.2056 


0670 14.9244 


0846 


11.8262 


0148 67.4019 
0151 66.1055 
0154 64.8580 
0157 63.6567 


0323 30.9599 
0326 30.6833 
0329 30.4116 
0332 30.1446 


0498 20.0872 
0501 19.9702 
0504 19.8546 
0507 19.7403 


0673 14.8596 
0676 14.7954 
0679 14.7317 
0682 14.6685 


0849 
0851 
0854 
0857 


11 . 7853 
11.7448 
11.7045 
11.6645 


0160 62.4992 


0335 29.8823 


0509 19.6273 


0685 14.6059 


0860 


11.6248 


0163 61.3829 
0166 60.3058 
0169 59.2659 
0172 58.2612 


0338 29.6245 
0340 29.3711 
0343 29.1220 
0346 28.8771 


0512 19.5156 
0515 19.4051 
0518 19.2959 
0521 19.1879 


0688 14.5438 
0690 14.4823 
0693 14.4212 
0696 14.3607 


0863 
0866 
0869 
0872 


11.5853 
11.5461 
11.5072 
11.4685 


0175 57.2900 


0349 28.6363 


0524 19.0811 


0699 14.3007 


0875 


11.4301 


cot tan 


cot tan 


cot tan 


cot tan 


cot 


tan 


r 


89° 


88° 


87° 


86° 


85° 


:.J 



NATURAL TANGENTS AND COTANGENTS 



83 



/ 


5° 


6° 


iyO 


8° 


9° 


/ 




1 

2 
3 
4 

5 

6 

7 
8 
9 

10 

11 
12 
13 

14 

15 

16 
17 
18 
19 

20 

21 
22 
23 

24 

25 

26 

27 
28 
29 

30 

31 
32 
33 
34 

35 

36 
37 
38 
39 

40 

41 
42 
43 
44 

45 

46 
47 
48 
49 

50 

51 

52 
53 
54 

55 

56 
57 
58 
59 

60 


tan 

0875 


cot 

11.4301 


tan 


cot 


tan 


cot 


tan 


cot 


tan 

1584 


cot 

6.3138 


60 

59 
58 

57 
56 

55 

54 
53 
52 
51 

50 

49 
48 
47 
46 

45 

44 
43 
42 
41 

40 

39 
38 
37 
36 

35 

34 
33 
32 
31 
30 
29 
28 
27 
26 

25 

24 
23 
22 
21 

20 

19 
18 
17 
16 

15 

14 
13 
12 
11 

10 

9 
8 
7 
6 
5 
4 
3 
2 
1 




1051 
1054 
1057 
1060 
1063 


9.5144 

9.4878 
9.4614 
9.4352 
9.4090 


1228 


8.1443 


1405 


7.1154 


0878 
0881 
0884 
0887 


11.3919 
11.3540 
11.3163 
11.2789 


1231 
1234 
1237 
1240 


8.1248 
8.1054 
8.0860 
8.0667 


1408 
1411 
1414 
1417 


7.1004 
7.0855 
7.0706 
7.0558 


1587 
1590 
1593 
1596 


6.3019 
6.2901 
6.2783 
6.2666 


0890 


11.2417 


1066 


9.3831 


1243 


8.0476 


1420 


7.0410 


1599 


6.2549 


0892 
0895 
0898 
0901 


11.2048 
11.1681 
11.1316 
11.0954 


1069 
1072 
1075 
1078 


9.3572 
9.3315 
9.3060 
9.2806 


1246 
1249 
1251 
1254 


8.0285 
8.0095 
7.9906 
7.9718 


1423 
1426 
1429 
1432 


7.0264 
7.0117 
6.9972 
6.9827 


1602 
1605 
1608 
1611 


6.2432 
6.2316 
6.2200 
6.2085 


0904 


11.0594 


1080 


9.2553 


1257 


7.9530 


1435 


6.9682 


1614 


6.1970 


0907 
0910 
0913 
0916 


11.0237 
10.9882 
10.9529 
10.9178 


1083 
1086 
1089 
1092 


9.2302 
9.2052 
9.1803 
9.1555 


1260 
1263 
1266 
1269 


7.9344 
7.9158 
7.8973 
7.8789 


1438 
1441 
1444 
1447 
1450 


6.9538 
6.9395 
6.9252 
6.9110 

6.8969 


1617 
1620 
1623 
1626 


6.1856 
6.1742 
6.1628 
6.1515 


0919 


10.8829 


1095 


9.1309 


1272 


7.8606 


1629 


6.1402 


0922 
0925 
0928 
0931 


10.8483 
10.8139 
10.7797 
10.7457 


1098 
1101 
1104 
1107 


9.1065 
9.0821 
9.0579 
9.0338 


1275 
1278 
1281 
1284 


7.8424 
7.8243 
7.8062 
7.7883 


1453 
1456 
1459 
1462 


6.8828 
6.8687 
6.8548 
6.8408 


1632 
1635 
1638 
1641 


6.1290 
6.1178 
6.1066 
6.0955 


0934 


10.7119 


1110 


9.0098 


1287 


7.7704 


1465 


6.8269 


1644 


6.0844 


0936 
0939 
0942 
0945 


10.6783 
10.6450 
10.6118 
10.5789 


1113 
1116 
1119 
1122 


8.9860 
8.9623 
8.9387 
8.9152 


1290 
1293 
1296 
1299 


7.7525 
7.7348 
7.7171 
7.6996 


1468 
1471 
1474 
1477 


6.8131 
6.7994 
6.7856 
6.7720 


1647 
1650 
1653 
1655 


6.0734 
6.0624 
6.0514 
6.0405 


0948 


10.5462 


1125 


8.8919 


1302 


7.6821 


1480 


6.7584 


1658 


6.0296 


0951 
0954 
0957 
0960 

0963 


10.5136 
10.4813 
10.4491 
10.4172 

10.3854 


1128 
1131 
1134 
1136 


8.8686 
8.8455 
8.8225 
8.7996 


1305 
1308 
1311 
1314 


7.6647 
7.6473 
7.6301 
7.6129 


1483 
1486 
1489 
1492 


6.7448 
6.7313 
6.7179 
6.7045 


1661 
1664 
1667 
1670 


6.0188 
6.0080 
5.9972 
5.9865 


1139 


8.7769 


1317 


7.5958 


1495 


6.6912 


1673 


5.9758 


0966 
0969 
0972 
0975 


10.3538 
10.3224 
10.2913 
10.2602 


1142 
1145 
1148 
1151 


8.7542 
8.7317 
8.7093 
8.6870 


1319 
1322 
1325 
1328 


7.5787 
7.5618 
7.5449 
7.5281 


1497 
1500 
1503 
1506 


6.6779 
6.6646 
6.6514 
6.6383 


1676 
1679 
1682 
1685 
1688 


5.9651 
5.9545 
5 . 9439 
5.9333 
5 . 9228 


0978 


10.2294 


1154 


8.6648 


1331 


7.5113 


1509 


6.6252 


0981 
0983 
0986 
0989 


10.1988 
10.1683 
10.1381 
10.1080 


1157 
1160 
1163 
1166 


8.6427 
8.6208 
8.5989 
8.5772 


1334 
1337 
1340 
1343 

1346 


7.4947 
7.4781 
7.4615 
7.4451 

7.4287 


1512 
1515 
1518 
1521 


6.6122 
6.5992 
6.5863 
6.5734 


1691 
1694 
1697 
1700 


5.9124 
5.9019 
5.8915 
5.8811 


0992 


10.0780 


1169 


8.5555 


1524 


6.5606 


1703 


5.8708 


0995 
0998 
1001 
1004 


10.0483 

10.0187 

9.9893 

9.9601 


1172 
1175 
1178 
1181 


8.5340 
8.5126 
8.4913 
8.4701 


1349 
1352 
1355 
1358 


7.4124 
7.3962 
7.3800 
7.3639 


1527 
1530 
1533 
1536 


6.5478 
6.5350 
6.5223 
6.5097 


1706 
1709 
1712 
1715 


5.8605 
5.8502 
5.8400 
5.8298 


1007 


9.9310 


1184 


8.4490 


1361 


7.3479 


1539 


6.4971 


1718 


5.8197 


1010 
1013 
1016 
1019 


9.9021 
9.8734 
9.8448 
9.8164 


1187 
1189 
1192 
1195 


8.4280 
8.4071 
8.3863 
8.3656 


1364 
1367 
1370 
1373 


7.3319 
7.3160 
7.3002 

7.2844 


1542 

1545 
1548 
1551 


6.4846 
6.4721 
6.4596 
6.4472 


1721 
1724 
1727 
1730 


5.8095 
5.7994 
5.7894 
5.7794 


1022 

1025 
1028 
1030 
1033 


9.7882 

9.7601 
9.7322 
9.7044 
9.6768 


1198- 


8.3450 


1376 


7.2687 


1554 


6.4348 


1733 


5.7694 


1201 
1204 
1207 
1210 


8.3245 
8.3041 
8.2838 
8.2636 


1379 
1382 
1385 
1388 


7.2531 
7.2375 
7.2220 
7.2066 


1557 
1560 
1563 
1566 


6.4225 
6.4103 
6.3980 
6.3859 


1736 
1739 
1742 
1745 


5 . 7594 
5.7495 
5 . 7396 
5.7297 


1036 


9.6499 


1213 


8.2434 


1391 


7.1912 


1569 


6.3737 


1748 


5.7199 


1039 
1042 
1045 
1048 


9.6220 

9.5949 

■ 9.5679 

9.5411 


1216 
1219 
1222 
1225 


8.2234 
8.2035 
8.1837 
8.1640 


1394 
1397 
1399 
1402 


7.1759 
7.1607 
7.1455 
7.1304 


1572 
1575 
1578 
1581 
1584 


6.3617 
6.3496 
6.3376 
6.3257 
6.3138 


1751 
1754 
1757 
1760 


5.7101 
5 . 7004 
5.6906 
5.6809 


1051 


9.5144 


1228 


8.1443 


1405 


7.1154 


1763 


5.6713 


cot 


tan 


cot 


tan 


cot 


tan 


cot 


tan 


cot 


tan 


1 


84° 


83° 


82° 


81° 


80° 





8 4 



NATURAL TANGENTS AND COTANGENTS 



/ 


10° 


11° 


12° 


13° 


14° 


/ 




1 

2 
3 
4 

5 

6 

7 
8 
9 

10 

11 
12 
13 
14 

15 

16 
17 
18 
19 

20 

21 
22 
23 
24 

25 

26 
27 
28 
29 

30 

31 
32 
33 
34 

35 

36 
37 
38 
39 

40 

41 
42 
43 
44 

45 

46 
47 
48 
49 

50 

51 
52 
53 
54 

55 

56 
57 
58 
59 

60 


tan 


cot 


tan 


cot 


tan 


cot 


tan 


cot 


tan 


cot 


60 

59 
58 
57 
56 

55 

54 
53 
52 
51 

50 

49 
48 
47 
46 

45 

44 
43 
42 
41 

40 

39 
38 
37 
36 

35 

34 
33 
32 
31 
30 
29 
28 
27 
26 

25 

24 
23 
22 
21 

20 

19 
18 
17 
16 

15 

14 
13 
12 
11 

10 

9 
8 
7 
6 
5 
4 
3 
2 
1 



1763 

1766 
1769 
1772 
1775 


5.6713 

5.6617 
5.6521 
5.6425 
5.6330 


1944 


5.1446 


2126 


4.7046 


2309 


4.3315 


2493 


4.0108 


1947 
1950 
1953 
1956 


5.1366 
5.1286 
5.1207 
5.1128 


2129 
2132 
2135 
2138 


4.6979 
4.6912 
4.6845 
4.6779 


2312 
2315 
2318 
2321 


4.3257 
4.3200 
4.3143 
4.3086 


2496 
2499 
2503 
2506 


4.0058 
4.0009 
3.9959 
3.9910 


1778 


5 . 6234 


1959 


5.1049 


2141 


4.6712 


2324 


4.3029 


2509 


3.9861 


1781 
1784 

1787 
1790 


5.6140 
5.6045 
5.5951 
5.5857 


1962 
1965 
1968 
1971 


5.0970 
5.0892 
5.0814 
5.0736 


2144 
2147 
2150 
2153 


4.6646 
4.6580 
4.6514 
4.6448 


2327 
2330 
2333 
2336 


4.2972 
4.2916 
4.2859 
4.2803 


2512 
2515 
2518 
2521 


3.9812 
3.9763 
3.9714 
3.9665 


1793 

1796 
1799 
1802 
1805 


5.5764 

5.5671 
5.5578 
5.5485 
5.5393 


1974 

1977 
1980 
1983 
1986 


5.0658 


2156 


4.6382 


2339 


4.2747 


2524 


3.9617 


5.0581 
5.0504 
5.0427 
5.0350 


2159 
2162 
2165 
2168 


4.6317 
4.6252 
4.6187 
4.6122 


2342 
2345 
2349 
2352 


4.2691 
4.2635 
4.2580 
4.2524 


2527 
2530 
2533 
2537 


3.9568 
3.9520 
3.9471 
3.9423 


1808 


5.5301 


1989 


5.0273 


2171 


4.6057 


2355 


4.2468 


2540 


3.9375 


1811 
1814 
1817 
1820 


5.5209 
5.5118 
5.5026 
5.4936 


1992 
1995 
1998 
2001 


5.0197 
5.0121 
5.0045 
4.9969 


2174 
2177 
2180 
2183 


4.5993 
4.5928 
4.5864 
4.5800 


2358 
2361 
2364 
2367 


4.2413 
4.2358 
4.2303 
4.2248 


2543 
2546 
2549 
2552 


3.9327 
3.9279 
3.9232 
3.9184 


1823 


5.4845 


2004 


4.9894 


2186 


4.5736 


2370 


4.2193 


2555 


3.9136 


1826 
1829 
1832 
1835 


5.4755 
5.4665 
5.4575 
5.4486 


2007 
2010 
2013 
2016 


4.9819 
4.9744 
4.9669 
4.9594 


2189 
2193 
2196 
2199 


4.5673 
4.5609 
4.5546 
4.5483 


2373 
2376 
2379 
2382 


4.2139 
4.2084 
4.2030 
4.1976 


2558 
2561 
2564 
2568 


3.9089 
3.9042 
3.8995 
3.8947 


1838 


5.4397 


2019 
2022 
2025 
2028 
2031 


4.9520 

4.9446 
4.9372 
4.9298 
4.9225 


2202 


4.5420 


2385 


4.1922 


2571 


3.8900 


1841 
1844 
1847 
1850 


5.4308 
5.4219 
5.4131 
5.4043 


2205 
2208 
2211 
2214 


4.5357 
4.5294 
4.5232 
4.5169 


2388 
2392 
2395 
2398 


4.1868 
4.1814 
4.1760 
4.1706 


2574 
2577 
2580 
2583 


3.8854 
3.8807 
3.8760 
3.8714 


1853 

1856 
1859 
1862 
1865 


5.3955 
5.3868 
5.3781 
5.3694 
5.3607 


2035 


4.9152 


2217 


4.5107 


2401 


4.1653 


2586 


3.8667 


2038 
2041 
2044 
2047 


4.9078 
4. "9006 
4.8933 
4.8860 


2220 
2223 
2226 
2229 


4.5045 
4.4983 
4.4922 
4.4860 


2404 
2407 
2410 
2413 


4.1600 
4.1547 
4.1493 
4.1441 


2589 
2592 
2595 
2599 


3.8621 
3.8575 
3.8528 
3.8482 


1868 


5.3521 


2050 


4.8788 


2232 


4.4799 


2416 


4.1388 


2602 


3.8436 


1871 
1874 

1877 
1880 


5.3435 
5.3349 
5.3263 
5.3178 


2053 
2056 
2059 
2062 


4.8716 
4.8644 
4.8573 
4.8501 


2235 
2238 
2241 
2244 


4.4737 
4.4676 
4.4615 
4.4555 


2419 
2422 
2425 
2428 


4.1335 
4.1282 
4.1230 
4.1178 


2605 
2608 
2611 
2614 


3.8391 
3.8345 
3.8299 
3.8254 


1883 


5.3093 


2065 


4.8430 


2247 


4.4494 


2432 


4.1126 


2617 


3.8208 


1887 
1890 
1893 
1896 


5.3008 
5.2924 
5.2839 
5.2755 


2068 
2071 
2074 
2077 


4.8359 
4.8288 
4.8218 
4.8147 


2251 
2254 
2257 
2260 


4.4434 
4.4374 
4.4313 
4.4253 


2435 
2438 
2441 
2444 


4.1074 
4.1022 
4.0970 
4.0918 


2620 
2623 
2627 
2630 


3.8163 
3.8118 
3.8073 
3.8028 


1899 


5.2672 


2080 


4.8077 


2263 


4.4194 


2447 


4.0867 


2633 


3.7983 


1902 
1905 
1908 
1911 


5.2588 
5.2505 
5.2422 
5.2339 


2083 
2086 
2089 
2092 


4.8007 
4.7937 
4.7867 
4.7798 


2266 
2269 
2272 
2275 


4.4134 
4.4075 
4.4015 
4.3956 


2450 
2453 
2456 
2459 


4.0815 
4.0764 
4.0713 
4.0662 


2636 
2639 
2642 
2645 


3.7938 
3.7893 
3.7848 
3.7804 


1914 


5.2257 


2095 


4.7729 


2278 


4.3897 


2462 


4.0611 


2648 


3.7760 


1917 
1920 
1923 
1926 

1929 


5.2174 
5.2092 
5.2011 
5.1929 


2098 
2101 
2104 
2107 


4.7659 
4.7591 
4.7522 
4.7453 


2281 
2284 
2287 
2290 


4.3838 
4.3779 
4.3721 
4.3662 


2465 
2469 
2472 
2475 


4.0560 
4.0509 
4.0459 
4.0408 


2651 
2655 
2658 
2661 


3.7715 
3 7671 
3.7627 
3.7583 


5.1848 


2110 


4.7385 


2293 


4.3604 


2478 


4.0358 


2664 


3.7539 


1932 
1935 
1938 
1941 


5.1767 
5.1686 
5.1606 
5.1526 


2113 
2116 
2119 
2123 


4.7317 
4.7249 
4.7181 
4.7114 


2296 
2299 
2303 
2306 


4.3546 
4.3488 
4.3430 
4.3372 


2481 
2484 
2487 
2490 


4.0308 
4.0257 
4.0207 
4.0158 


2667 
2670 
2673 
2676 


3.7495 
3.7451 
3 . 7408 
3 . 7364 


1944 


5.1446 


2126 


4 . 7046 


2309 


4.3315 


2493 


4.0108 


2679 


3 7321 


cot 


tan 


cot 


tan 


cot 


tan 


cot 


tan 


cot 


tan 


/ 


79° 


78° 


rymO 


76° 


75° 


/ 



NATURAL TANGENTS AND COTANGENTS 



85 



r 


15° 


16° 


17° 


18° 


19° 


/ 




1 

2 
3 
4 

5 

6 

7 
8 
9 

10 

11 

12 
13 
14 

15 

16 
17 
18 
19 

20 

21 
22 
23 
24 

25 

26 
27 
28 
29 

30 

31 
32 
33 
34 

35 

36 
37 
38 
39 

40 

41 
42 
43 

44 

45 

46 
47 
48 
49 

50 

51 
52 
53 

54 

55 

56 
57 
58 
59 

60 


tan 


cot 


tan 


cot 


tan 


cot 


tan 


cot 


tan 


cot 


60 

59 
58 
57 
56 

55 

54 
53 
52 
51 

50 

49 
48 
47 
46 

45 

44 
43 
42 
41 

40 

39 
38 
37 
36 

35 

34 
33 
32 
31 
30 
29 
28 
27 
26 

25 

24 
23 
22 
21 

20 

19 
18 
17 
16 

15 

14 
13 
12 
11 

10 

9 
8 
7 
6 
5 
4 
3 
2 
1 




2679 


3.7321 


2867 


3.4874 


3057 


3.2709 


3249 


3.0777 


3443 


2.9042 


2683 
2686 
2689 
2692 


3.7277 
3.7234 
3.7191 
3.7148 


2871 
2874 
2877 
2880 


3.4836 
3.4798 
3.4760 
3.4722 


3060 
3064 
3067 
3070 


3.2675 
3.2641 
3.2607 
3.2573 


3252 
3256 
3259 
3262 


3.0746 
3.0716 
3.0686 
3.0655 


3447 
3450 
3453 
3456 


2.9015 
2.8987 
2.8960 
2.8933 


2695 


3.7105 


2883 


3.4684 


3073 


3.2539 


3265 


3.0625 


3460 


2.8905 


2698 
2701 
2704 
2708 


3.7062 
3.7019 
3.6976 
3.6933 


2886 
2890 
2893 
2896 


3.4646 
3.4608 
3.4570 
3.4533 


3076 

3080 

3083 

. 3086 


3.2506 
3.2472 
3.2438 
3.2405 


3269 
3272 
3275 
3278 


3.0595 
3.0565 
3.0535 
3.0505 


3463 
3466 
3469 
3473 


2.8878 
2.8851 
2.8824 
2.8797 


2711 


3.6891 


2899 


3.4495 


3089 


3.2371 


3281 


3.0475 


3476 


2.8770 


2714 
2717 
2720 
2723 

2726 


3.6848 
3.6806 
3.6764 
3.6722 

3.6680 


2902 
2905 
2908 
2912 


3.4458 
3.4420 
3.4383 
3.4346 


3092 
3096 
3099 
3102 


3.2338 
3.2305 
3.2272 
3.2238 


3285 
3288 
3291 
3294 


3.0445 
3.0415 
3.0385 
3.0356 


3479 
3482 
3486 
3489 


2.8743 
2.8716 
2.8689 
2.8662 


2915 


3.4308 


3105 


3.2205 


3298 


3.0326 


3492 


2.8636 


2729 
2733 
2736 
2739 


3.6638 
3.6596 
3.6554 
3.6512 


2918 
2921 
2924 
2927 


3.4271 
3.4234 
3.4197 
3.4160 


3108 
3111 
3115 
3118 


3.2172 
3.2139 
3.2106 
3.2073 


3301 
3304 
3307 
3310 


3.0296 
3.0267 
3.0237 
3.0208 


3495 
3499 
3502 
3505 


2.8609 
2.8582 
2.8556 
2.8529 


2742 


3.6470 


2931 


3.4124 


3121 


3.2041 


3314 


3.0178 


3508 


2.8502 


2745 
2748 
2751 
2754 


3.6429 
3.6387 
3.6346 
3.6305 


2934 
2937 
2940 
2943 


3.4087 
3.4050 
3.4014 
3.3977 


3124 
3127 
3131 
3134 


3.2008 
3.1975 
3.1943 
3.1910 


3317 
3320 
3323 
3327 


3.0149 
3.0120 
3.0090 
3.0061 


3512 
3515 
3518 
3522 


2.8476 
2.8449 
2.8423 
2.8397 


2758 


3.6264 


2946 

2949 
2953 
2956 
2959 


3.3941 


3137 


3.1878 


3330 


3.0032 


3525 


2.8370 


2761 
2764 
2767 
2770 


3.6222 
3.6181 
3.6140 
3.6100 


3.3904 
3.3868 
3.3832 
3.3796 


3140 
3143 

3147 
3150 


3.1845 
3.1813 

3.1780 
3.1748 


3333 
3336 
3339 
3343 


3.0003 
2.9974 
2.9945 
2.9916 


3528 
3531 
3535 
3538 
3541 


2.8344 
2.8318 
2.8291 
2.8265 

2.8239 


2773 


3.6059 


2962 


3.3759 


3153 


3.1716 


3346 


2.9887 


2776 
2780 
2783 
2786 


3.6018 
3.5978 
3.5937 
3.5897 


2965 
2968 
2972 
2975 


3.3723 
3.3687 
3.3652 
3.3616 


3156 
3159 
3163 
3166 


3.1684 
3.1652 
3.1620 
3.1588 


3349 
3352 
3356 
3359 


2.9858 
2.9829 
2.9800 
2.9772 


3544 
3548 
3551 
3554 


2.8213 

2.8187 
2.8161 
2.8135 


2789 


3.5856 


2978 


3.3580 


3169 


3.1556 


3362 


2.9743 


3558 


2.8109 


2792 
2795 
2798 
2801 


3.5816 
3.5776 
3.5736 
3.5696 


2981 
2984 
2987 
2991 


3.3544 
3.3509 
3.3473 
3.3438 


3172 
3175 
3179 
3182 


3.1524 
3.1492 
3.1460 
3.1429 


3365 
3369 
3372 
3375 


2.9714 
2.9686 
2.9657 
2.9629 


3561 
3564 
3567 
3571 


2.8083 
2.8057 
2.8032 
2.8006 


2805 


3.5656 


2994 


3.3402 


3185 


3.1397 


3378 


2.9600 


3574 


2.7980 


2808 
2811 
2814 
2817 


3.5616 
3.5576 
3.5536 
3.5497 


2997 
3000 
3003 
3006 


3.3367 
3.3332 
3.3297 
3.3261 


3188 
3191 
3195 
3198 


3.1366 
3.1334 
3.1303 
3.1271 


3382 
3385 
3388 
3391 


2.9572 
2.9544 
2.9515 
2.9487 


3577 
3581 
3584 
3587 


2.7955 
2 . 7929 
2.7903 
2.7878 


2820 


3.5457 


3010 


3.3226 


3201 


3.1240 


3395 


2.9459 


3590 


2.7852 


2823 
2827 
2830 
2833 


3.5418 
3.5379 
3.5339 
3.5300 


3013 
3016 
3019 
3022 


3.3191 
3.3156 
3.3122 
3.3087 


3204 
3207 
3211 
3214 


3.1209 
3.1178 
3.1146 
3.1115 


3398 
3401 
3404 
3408 


2.9431 
2.9403 
2.9375 
2.9347 


3594 
3597 
3600 
3604 

3607 


2.7827 
2.7801 
2.7776 
2.7751 

2.7725 


2836 


3.5261 


3026 


3.3052 


3217 


3.1084 


3411 


2.9319 


2839 
2842 
2845 
2849 


3.5222 
3.5183 
3.5144 
3.5105 


3029 
3032 
3035 
3038 


3.3017 
3.2983 
3.2948 
3.2914 


3220 
3223 
3227 
3230 


3 . 1053 
3.1022 
3.0991 
3.0961 


3414 
3417 
3421 
3424 


2.9291 
2.9263 
2.9235 
2.9208 


3610 
3613 
3617 
3620 


2.7700 
2.7675 
2.7650 
2.7625 


2852 


3.5067 


3041 


3.2880 


3233 


3.0930 


3427 


2.9180 


3623 


2.7600 


2855 
2858 
2861 
2864 

2867 


3.5028 
3.4989 
3.4951 
3.4912 

3.4874 


3045 
3048 
3051 
3054 


3.2845 
3.2811 
3.2777 
3.2743 


3236 
3240 
3243 
3246 


3.0899 
3.0868 
3.0838 
3.0807 


3430 
3434 
3437 
3440 


2.9152 
2.9125 
2.9097 
2.9070 


3627 
3630 
3633 
3636 


2.7575 
2.7550 
2.7525 
2.7500 


3057 


3.2709 


3249 


3.0777 


3443 


2.9042 


3640 
cot 


2.7475 
tan 


cot 


tan 


cot 


tan 


cot 


tan 


cot 


tan 


i 


74° 


73° 


72° 


71° 


70° 


/ 



86 



NATURAL TANGENTS AND COTANGENTS 



/ 


20° 


21° 


22° 


23° 


24° 


I 




1 

2 
3 
4 

5 

6 

7 
8 
9 

10 

11 

12 
13 
14 

15 

16 
17 

18 
19 

20 

21 
22 
23 

24 

25 

26 

27 
28 
29 

30 

31 
32 
33 
34 

35 

36 
37 
38 
39 
40 
41 
42 
43 
44 

45 

46 

47 
48 
49 

50 

51 
52 
53 
54 

55 

56 
57 
58 
59 

60 


tan 


cot 


tan 


cot 


tan 


cot 


tan 


cot 


tan 


cot 


60 

59 
58 

57 
56 

55 

54 
53 
52 
51 
50 
49 
48 
47 
46 

45 

44 
43 
42 
41 

40 

39 
38 
37 
36 

35 

34 
33 
32 
31 

30 

29 
28 
27 
26 

25 

24 
23 
22 
21 

20 

19 
18 
17 
16 

15 

14 
13 
12 
11 

10 

9 
8 

7 
6 

5 

4 
3 
2 
1 




3640 


2.7475 


3839 


2.6051 


4040 


2.4751 


4245 


2.3559 


4452 


2.2460 


3643 
3646 
3650 
3653 


2.7450 
2.7425 
2.7400 
2.7376 


3842 
3845 
3849 
3852 


2.6028 
2.6006 
2.5983 
2.5961 


4044 
4047 
4050 
4054 


2.4730 
2.4709 
2.4689 
2.4668 


4248 
4252 
4255 
4258 


2.3539 
2.3520 
2.3501 
2.3483 


4456 
4459 
4463 
4466 


2.2443 
2.2425 
2.2408 
2.2390 


3656 


2.7351 


3855 


2.5938 


4057 


2.4648 


4262 


2.3464 


4470 


2.2373 


3659 
3663 
3666 
3669 

3673 

3676 
3679 
3683 
3686 


2.7326 
2.7302 
2.7277 
2.7253 

2.7228 

2.7204 
2.7179 
2.7155 
2.7130 


3859 
3862 
3865 
3869 


2.5916 
2.5893 
2.5871 
2.5848 


4061 
4064 
4067 
4071 


2.4627 
2.4606 
2.4586 
2.4566 


4265 
4269 
4272 
4276 


2.3445 
2.3426 
2.3407 
2.3388 


4473 
4477 
4480 
4484 


2.2355 
2.2338 
2.2320 
2.2303 


3872 


2.5826 


4074 


2.4545 


4279 


2.3369 


4487 


2.2286 


3875 
3879 

3882 
3885 


2.5804 
2.5782 
2.5759 
2.5737 


4078 
4081 
4084 
4088 


2.4525 
2.4504 
2.4484 
2.4464 


4283 
4286 
4289 
4293 


2.3351 
2.3332 
2.3313 
2.3294 


4491 
4494 
4498 
4501 


2.2268 
2.2251 
2.2234 
2.2216 


3689 

3693 
3696 
3699 
3702 


2.7106 
2.7082 
2.7058 
2.7034 
2.7009 


3889 


2.5715 


4091 


2.4443 


4296 


2.3276 


4505 


2.2199 


3892 
3895 
3899 
3902 


2.5693 
2.5671 
2.5649 
2.5627 


4095 
4098 
4101 
4105 


2.4423 
2.4403 
2.4383 
2.4362 


4300 
4303 
4307 
4310 


2.3257 
2.3238 
2.3220 
2.3201 


4508 
4512 
4515 
4519 


2.2182 
2.2165 
2.2148 
3.2130 


3706 


2.6985 


3906 


2.5605 


4108 


2.4342 


4314 


2.3183 


4522 


2.2113 


3709 
3712 
3716 
3719 


2.6961 
2.6937 
2.6913 
2.6889 


3909 
3912 
3916 
3919 


2.5533 
2.5561 
2.5539 
2.5517 


4111 
4115 
4118 
4122 


2.4322 
2.4302 
2.4282 
2.4262 


4317 
4320 
4324 
4327 


2.3164 
2.3146 
2.3127 
2.3109 


4526 
4529 
4533 
4536 
4540 


2.2096 
2.2079 
2.2062 
2.2045 

2.2028 


3722 


2.6865 


3922 


2.5495 


4125 


2.4242 


4331 


2.3090 


3726 
3729 
3732 
3736 


2.6841 
2.6818 
2.6794 
2.6770 


3926 
3929 
3932 
3936 


2.5473 
2.5452 
2.5430 
2.5408 


4129 
4132 
4135 
4139 


2.4222 
2.4202 
2.4182 
2.4162 


4334 
4338 
4341 
4345 


2.3072 
2.3053 
2.3035 
2.3017 


4543 
4547 
4550 
4554 


2.2011 
2.1994 
2.1977 
2.1960 


3739 
3742 
3745 
3749 
3752 


2.6746 
2.6723 
2.6699 
2.6675 
2.6652 


3939 


2.5386. 


4142 


2.4142 


4348 


2.2998 


4557 


2 . 1943 


3942 
3946 
3949 
3953 


2.5365 
2.5343 
2.5322 
2.5300 


4146 
4149 
4152 
4156 


2.4122 
2.4102 
2.4083 
2.4063 


4352 
4355 
4359 
4362 


2.2980 
2.2962 
2.2944 
2.2925 


4561 
4564 
4568 
4571 


2.1926 
2.1909 
2.1892 
2.1876 


3755 


2.6628 


3956 


2.5279 


4159 


2.4043 


4365 


2.2907 


4575 


2.1859 


3759 
3762 
3765 
3769 


2.6605 
2.6581 
2.6558 
2.6534 


3959 
3963 
3966 
3969 


2.5257 
2.5236 
2.5214 
2.5193 


4163 
4166 
4169 
4173 


2.4023 
2.4004 
2.3984 
2.3964 


4369 
4372 
4376 
4379 


2.2889 
2.2871 
2.2853 
2.2835 


4578 
4582 
4585 
4589 


2.1842 
2.1825 
2.1808 
2.1792 


3772 


2.6511 


3973 


2.5172 


4176 


2.3945 


4383 


2.2817 


4592 


2.1775 


3775 
3779 
3782 
3785 


2.6488 
2.6464 
2.6441 
2.6418 


3976 
3979 
3983 
3986 


2.5150 
2.5129 
2.5108 
2.5086 


4180 
4183 
4187 
4190 


2.3925 
2.3906 
2.3886 
2.3867 


4386 
4390 
4393 
4397 


2.2799 
2.2781 
2.2763 
2.2745 


4596 
4599 
4603 
4607 

4610 


2.1758 
2.1742 
2.1725 
2.1708 
2.1692 


3789 


2.63J5 


3990 


2.5065 


4193 


2.3847 


4400 


2.2727 


3792 
3795 
3799 
3802 


2.6371 
2.6348 
2.6325 
2.6302 


3993 
3996 
4000 
4003 


2.5044 
2.5023 
2.5002 
2.4981 


4197 
4200 
4204 
4207 


2.3828 
2.3808 
2.3789 
2.3770 


4404 
4407 
4411 
4414 


2.2709 
2.2691 
2.2673 
2.2655 


4614 
4617 
4621 
4624 


2.1675 
2.1659 
2.1642 
2.1625 


3805 

3809 
3812 
3815 
3819 


2.6279 

2.6256 
2.6233 
2.6210 
2.6187 


4006 


2.4960 


4210 


2.3750 


4417 


2.2637 


4628 


2.1609 


4010 
4013 
4017 
4020 


2.4939 
2.4918 

2.4897 
2.4876 


4214 
4217 
4221 
4224 


2.3731 
2.3712 
2.3693 
2.3673 


4421 

4424 
4428 
4431 


2.2620 
2.2602 
2.2584 
2.2566 


4631 
4635 
4638 
4642 


2.1592 
2.1576 
2.1560 
2.1543 


3822 2 . 6165 

3825 2.6142 
3829 2.6119 
3832 2.6096 
3835 2 . 6074 

3839 2.6051 
cot tan 

69° 


4023 


2.4855 


4228 


2.3654 


4435 


2.2549 


4645 


2.1527 


4027 
4030 
4033 
4037 


2.4834 
2.4813 
2.4792 

2.4772 


4231 
4234 
4238 
4241 


2.3635 
2.3616 
2.3597 
2.3578 


4438 
4442 
4445 
4449 


2.2531 
2.2513 
2.2496 
2.2478 


4649 
4652 
4656 
4660 


2.1510 
2 . 1494 
2 . 1478 
2.1461 


4040 


2.4751 


4245 


2.3559 


4452 


2.2460 


4663 


2 . 1445 




cot 


tan 


cot 


tan 


cot 


tan 


cot tan 
65° 


1 


68° 


67° 


66° 


/ 



NATURAL TANGENTS AND COTANGENTS 



87 



/ 


2 


5° 


26° 


27° 


28° 


29° 


/ 




1 

2 
3 

4 

5 

6 

7 
8 
9 

10 

11 

12 
13 
14 

15 

16 

17 
18 
19 

20 

21 
22 
23 

24 

25 

26 

27 
28 
29 

30 

31 
32 
33 
34 

35 

36 
37 
38 
39 

40 

41 
42 
43 

44 

45 

46 
47 
48 

49 

50 

51 
52 
53 
54 

55 

56 
57 

58 
59 

60 


tan 


cot 


tan 


cot 


tan 


cot 


tan 


cot 


tan 


cot 


60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 
38 
37 
36 

35 

34 
33 
32 
31 
30 
29 
28 
27 
26 

25 

24 
23 
22 
21 

20 

19 
18 
17 
16 

15 

14 
13 
12 
11 

10 

9 

8 
7 
6 

5 

4 
3 
2 
1 




4663 


2.1445 


4877 
4881 
4885 
4888 
4892 


2.0503 

2.0488 
2.0473 
2.0458 
2.0443 


5095 


1.9626 


5317 


1.8807 


5543 


1.8040 


4667 
4670 
4674 
4677 


2.1429 
2.1413 
2.1396 
2.1380 


5099 
5103 
5106 
5110 


1.9612 
1.9598 
1.9584 
1.9570 


5321 
5325 
5328 
5332 


1.8794 
1.8781 
1.8768 
1.8755 


5547 
5551 
5555 
5558 


1.8028 
1.8016 
1.8003 
1.7991 


4681 


2.1364 


4895 


2.0428 


5114 


1.9556 


5336 


1.8741 


5562 


1.7979 


4684 
4688 
4691 
4695 


2.1348 
2 . 1332 
2.1315 
2.1299 


4899 
4903 
4906 
4910 

4913 


2.0413 
2.0398 
2.0383 
2.0368 

2.0353 


5117 
5121 
5125 
5128 


1.9542 
1.9528 
1.9514 
1.9500 


5340 
5343 
5347 
5351 


1.8728 
1.8715 
1.8702 
1.8689 


5566 
5570 
5574 
5577 


1.7966 
1 . 7954 
1 . 7942 
1.7930 


4699 


2.1283 


5132 


1 . 9486 


5354 


1.8676 


5581 


1.7917 


4702 
4706 
4709 
4713 


2.1267 
2.1251 
2.1235 
2.1219 


4917 
4921 
4924 
4928 


2.0338 
2.0323 
2.0308 
2.0293 


5136 
5139 
5143 
5147 


1.9472 
1.9458 
1.9444 
1.9430 


5358 
5362 
5366 
5369 


1.8663 
1.8650 
1.8637 
1.8624 


5585 
5589 
5593 
5596 


1.7905 
1 . 7893 
1.7881 
1.7868 


4716 


2.1203 


4931 


2.0278 


5150 


1.9416 


5373 


1.8611 


5600 


1.7856 


4720 
4723 
4727 
4731 


2.1187 
2.1171 
2.1155 
2.1139 


4935 
4939 
4942 
4946 


2.0263 
2.0248 
2.0233 
2.0219 


5154 
5158 
5161 
5165 


1.9402 
1.9388 
1.9375 
1.9361 


5377 
5381 
5384 
5388 


1.8598 
1.8585 
1.8572 
1.8559 


5604 
5608 
5612 
5616 


1 . 7844 
1.7832 
1 . 7820 
1 . 7808 


4734 


2.1123 


4950 


2.0204 


5169 


1.9347 


5392 


1.8546 


5619 


1 . 7796 


4738 
4741 
4745 
4748 


2.1107 
2.1092 
2.1076 
2.1060 


4953 
4957 
4960 
4964 


2.0189 
2.0174 
2.0160 
2.0145 


5172 
5176 
5180 
5184 


1.9333 
1.9319 
1.9306 
1.9292 


5396 
5399 
5403 
5407 


1.8533 
1.8520 
1.8507 
1.8495 


5623 
5627 
5631 
5635 


1.7783 
1.7771 
1.7759 

1.7747 


4752 


2.1044 


4968 


2.0130 


5187 


1.9278 


5411 


1.8482 


5639 


1.7735 


4755 
4759 
4763 
4766 


2.1028 
2.1013 

2.0997 
2.0981 


4971 
4975 
4979 
4982 


2.0115 
2.0101 
2.0086 
2.0072 


5191 
5195 

5198 
5202 

5206 


1.9265 
1.9251 
1.9237 
1.9223 

1.9210 


5415 
5418 
5422 
5426 


1.8469 
1.8456 
1.8443 
1.8430 


5642 
5646 
5650 
5654 


1 . 7723 
1.7711 
1.7699 
1 . 7687 


4770 


2.0965 


4986 


2.0057 


5430 


1.8418 


5658 


1.7675 


4773 
4777 
4780 
4784 


2.0950 
2.0934 ' 
2.0918 
2.0903 


4989 
4993 
4997 
5000 


2.0042 
2.0028 
2.0013 
1.9999 


5209 
5213 
5217 
5220 


1.9196 
1.9183 
1.9169 
1.9155 


5433 
5437 
5441 
5445 


1.8405 
1.8392 
1.8379 
1.8367 


5662 
5665 
5669 
5673 


1 . 7663 
1.7651 
1 . 7639 
1 . 7627 


4788 


2.0887 


5004 


1.9984 


5224 


1.9142 


5448 


1.8354 


5677 


1.7615 


4791 
4795 
4798 
4802 


2.0872 
2.0856 
2.0840 
2.0825 


5008 
5011 
5015 
5019 

5022 


1.9970 
1.9955 
1.9941 
1.9926 
1.9912 


5228 
5232 
5235 
5239 


1.9128 
1.9115 
1.9101 
1.9088 


5452 
5456 
5460 
5464 


1.8341 
1.8329 
1.8316 
1.8303 


5681 
5685 
5688 
5692 


1.7603 
1.7591 
1 . 7579 
1 . 7567 


4806 


2.0809 


5243 


1.9074 


5467 


1.8291 


5696 


1.7556 


4809 
4813 
4816 
4820 


2.0794 
2.0778 
2.0763 
2.0748 


5026 
5029 
5033 
5037 


1.9897 
1.9883 
1.9868 
1.9854 


5246 
5250 
5254 
5258 


1.9061 
1.9047 
1.9034 
1.9020 


5471 
5475 
5479 
5482 


1.8278 
1.8265 
1.8253 
1.8240 


5700 
5704 
5708 
5712 

5715 


1.7544 
1.7532 
1.7520 
1.7508 

1.7496 


4823 

4827 
4831 
4834 
4838 


2.0732 


5040 


1.9840 


5261 


1.9007 


5486 


1.8228 


2.0717 
2.0701 
2 . 0686 
2.0671 


5044 
5048 
5051 
5055 


1.9825 
1.9811 
1.9797 
1.9782 


5265 
5269 
5272 
5276 
5280 
5284 
5287 
5291 
5295 

5298 


1.8993 
1.8980 
1.8967 
1 8953 

1.8940 
1.8927 
1.8913 
1.8900 
1.8887 

1.8873 


5490 
5494 
5498 
5501 


1.8215 
1.8202 
1.8190 
1.8177 


5719 
5723 
5727 
5731 


1.7485 
1.7473 
1.7461 
1.7449 


4841 


2.0655 


5059 
5062 
5066 
5070 
5073 


1.9768 
1.9754 
1 . 9740 
1.9725 
1.9711 


5505 


1.8165 


5735 


1 . 7437 


4845 
4849 
4852 
4856 


2.0640 
2.0625 
2 . 0609 
2.0594 


5509 
5513 
5517 
5520 


1.8152 
1.8140 
1.8127 
1.8115 


5739 
5743 
5746 
5750 


1.7426 
1.7414 
1.7402 
1.7391 


4859 


2.0579 


5077 


1.9697 


5524 


1.8103 


5754 


1.7379 


4863 
4867 
4870 
4874 


2.0564 
2.0549 
2.0533 
2.0518 


5081 
5084 
5088 
5092 

5095 


1.9683 
1.9669 
1.9654 
1.9640 

1.9626 


5302 
5306 
5310 
5313 
5317 


1.8860 
1.8847 
1.8834 
1.8820 
1.8807 


5528 
5532 
5535 
5539 


1.8090 
1.8078 
1.8065 
1.8053 


5758 
5762 
5766 
5770 


1.7367 
1 . 7355 
1.7344 
1 . 7332 


4877 


2.0503 


5543 


1.8040 


5774 


1.7321 


cot 


tan 


cot 


tan 


cot 


tan 


cot 


tan 


cot 


tan 


/ 


64° 


63° 


62° 


61° 


60° 


/ 



88 



NATURAL TANGENTS AND COTANGENTS 



' 


30° 


31° 


33° 


33° 


34° 


/ 




1 

2 
3 
4 

5 

6 

7 
8 
9 

10 

11 

12 
13 

14 

15 

16 
17 
18 
19 

30 

21 
22 
23 
24 

35 

26 

27 

28 
29 

30 

31 
32 
33 
34 

35 

36 
37 
38 
39 

40 

41 
42 
43 
44 

45 

46 
47 
48 
49 

50 

51 
52 
53 
54 

55 

56 

57 
58 
59 

60 


tan 


cot 


tan 


cot 


tan 


cot 


tan 


cot 


tan 


cot 


60 

59 

58 
57 
56 

55 

54 
53 
52 
51 

50 

49 
48 
47 
46 

45 

44 
43 
42 
41 

40 

39 
38 
37 
36 

35 

34 
33 
32 
31 
30 
29 
28 
■ 27 
26 

35 

24 
23 
22 
21 

30 

19 
18 
17 
16 
15 
14 
13 
12 
11 

10 

9 

8 
7 
6 

5 

4 
3 
2 
1 




5774 


1.7321 


6009 


1.6643 


6249 


1.6003 


6494 


1.5399 


6745 


1.4826 


5777 
5781 
5785 
5789 


1.7309 
1.7297 
1.7286 
1.7274 


6013 
6017 
6020 
6024 


1.6632 
1.6621 
1.6610 
1.6599 


6253 
6257 
6261 
6265 


1.5993 
1.5983 
1.5972 
1.5962 


6498 
6502 
6506 
6511 


1.5389 
1.5379 
1.5369 
1.5359 


6749 
6754 
6758 
6762 


1.4816 
1.4807 
1.4798 
1.4788 


5793 


1.7262 


6028 


1.6588 


6269 


1.5952 


6515 


1.5350 


6766 


1.4779- 


5797 
5801 
5805 
5808 


1.7251 
1.7239 
1.7228 
1.7216 


6032 
6036 
6040 
6044 


1.6577 
1.6566 
1.6555 
1.6545 


6273 
6277 
6281 
6285 


1.5941 
1.5931 
1.5921 
1.5911 


6519 
6523 
6527 
6531 


1.5340 
1.5330 
1.5320 
1.5311 


6771 
6775 
6779 
6783 


1.4770 
1.4761 
1.4751 
1.4742 


5812 


1.7205 


6048 


1.6534 


6289 


1.5900 


6536 


1.5301 


6787 


1.4733 


5816 
5820 

5824 
5828 


1.7193 
1.7182 
1.7170 
1.7159 


6052 
6056 
6060 
6064 


1.6523 
1.6512 
1.6501 
1.6490 


6293 
6297 
6301 
6305 


1.5890 
1.5880 
1.5869 
1.5859 


6540 
6544 
6548 
6552 


1.5291 
1.5282 
1.5272 
1.5262 


6792 
6796 
6800 
6805 


1.4724 
1.4715 
1.4705 
1.4696 


5832 


1.7147 


6068 


1.6479 


6310 


1.5849 


6556 


1.5253 


6809 


1.4687 


5836 
5840 
5844 

5847 


1.7136 
1.7124 
1.7113 
1.7102 


6072 
6076 
6080 
6084 


1.6469 
1.6458 
1.6447 
1.6436 


6314 
6318 
6322 
6326 


1.5839 
1.5829 
1.5818 
1.5808 


6560 
6565 
6569 
6573 


1.5243 
1.5233 
1.5224 
1.5214 


6813 
6817 
6822 
6826- 


1.4678 
1.4669 
1.4659 
1.4650 


5851 


1.7090 


6088 


1.6426 


6330 


1.5798 


6577 


1.5204 


6830 


1.4641 


5855 
5859 
5863 
5867 


1 . 7079 
1.7067 
1.7056 
1.7045 


6092 
6096 
6100 
6104 


1.6415 
1.6404 
1.6393 
1.6383 


6334 
6338 
6342 
6346 


1.5788 
1.5778 
1.5768 
1.5757 


6581 
6585 
6590 
6594 


1.5195 
1.5185 
1.5175 
1.5166 


6834 
6839 
6843 
6847 


1.4632 
1.4623 
1.4614 
1.4605 


5871 


1.7033 


6108 


1.6372 


6350 


1.5747 


6598 


1.5156 


6851 


1.4596 


5875 
5879 

5883 
5887 


1.7022 
1.7011 
1.6339 
1.6383 


6112 
6116 
6120 
6124 


1.6361 
1.6351 
1.6340 
1.6329 


6354 
6358 
6363 
6367 


1.5737 
1.572? 
1.5717 
1.5707 


6602 
6606 
6610 
6615 


1.5147 
1.5137 
1.5127 
1.5118 


6856 
6860 
6864 
6869 


1.4586 
1.4577 
1.4568 
1.4559 


5890 


1.6J/7 


6128 


1.6319 


6371 


1.5697 


6619 


1.5108 


6873 


1.4550 


5894 
5898 
5902 
5906 


1.6365 
1.6351 
1.6313 
1.6332 


6132 
6136 
6140 
6144 


1.6308 
1.6297 
1 . 6287 
1.6276 


6375 
6379 
6383 
6387 


1.5687 
1.5677 
1.5667 
1.5657 


6623 
6627 
6631 
6636 


1.5099 
1.5089 
1.5080 
1.5070 


6877 
6881 
6886 
6890 


1.4541 
1.4532 
1.4523 
1.4514 


5910 
5914 
5918 
5922 
5926 


1.6323 

1.6909 
1.6338 
1.6337 
1.6375 


6148 

6152 
6156 
6160 
6164 


1.6265 

1.6255 
1.6244 
1 . 6234 
1 . 6223 


6391 


1.5647 


6640 


1.5061 


6894 


1.4505 


6395 
6399 
6403 
6408 


1.5637 
1.5627 
1.5617 
1.5607 


6644 
6648 
6652 
6657 


1.5051 
1.5042 
1.5032 
1.5023 


6899 
6903 
6907 
6911 


1.4496 
1.4487 
1.4478 
1.4469 


5930 


1.6331 


6168 


1.6212 


6412 


1.5597 


6661 


1.5013 


6916 


1.4460 


5934 
5938 
5942 
5945 


1.6353 
1.6342 
1.6331 

1.6320 


6172 
6176 
6180 
6184 


1 . 6202 
1.6191 
1.6181 
1.6170 


6416 
6420 
6424 
6428 


1.5587 
1.5577 
1.5567 
1.5557 


6665 
6669 
6673 
6678 


1.5004 
1.4994 
1.4985 
1.4975 


6920 
6924 
6929 
6933 


1.4451 
1.4442 
1.4433 

1.4424 


5949 


1.6303 


6188 


1.6160 


6432 


1.5547 


6682 


1.4966 


6937 


1.4415 


5953 
5957 
5961 
5965 
5969 


1.6797 
1.6786 
1.6775 
1.6761 
1.6753 


6192 
6196 
6200 
6204 


1.6149 
1.6139 
1.6128 
1.6118 


6436 
6440 
6445 
6449 


1.5537 
1.5527 
1.5517 
1.5507 


6686 
6690 
6694 
6699 


1.4957 
1.4947 
1.4938 
1.4928 


6942 
6946 
6950 
6954 


1.4406 
1.4397 
1.4388 
1.4379 


6208 


1.6107 


6453 


1.5497 


6703 


1.4919 


6959 


1.4370 


5973 

5977 
5981 
5985 


1.6742 
1.6731 
1.6720 
1.6709 


6212 
6216 
6220 
6224 


1.6097 
1 . 6087 
1.6076 
1.6066 


6457 
6461 
6465 
6469 


1.5487 
1.5477 
1.5468 
1.5458 


6707 
6711 
6716 
6720 


1.4910 
1.4900 
1.4891 
1.4882 


6963 
6967 
6972 
6976 


1.4361 
1.4352 
1.4344 
1.4335 


5989 


1.6698 


6228 


1.6055 


6473 


1.5448 


6724 


1.4872 


6980 


1.4326 


5993 
5997 
6001 
6005 


1.6637 
1.6676 
1.6665 
1.6654 


6233 

6237 

6241 

. 6245 


1.6045 
1.6034 
1.6024 
1.6014 


6478 
6482 
6486 
6490 


1.5438 
1.5428 
1-.5418 
1.5408 


6728 
673-2 
6737 
6741 


1.4863 
1.4854 
1.4844 
1.4835 


6985 
6989 
6993 
6998 


1.4317 
1.4308 
1.4299 
1.4290 


6009 
cot 


1.6643 
tan 


6249 


1.6003 


6494 


1.5399 


6745 


1.4826 


7002 


1.4281 


cot tan 

58° 


cot 


tan 


cot 


tan 


cot 


tan 


t 


59° 


57° 


56° 


55° 


t 



NATURAL TANGENTS AND COTANGENTS 



89 



/ 


35° 


36° 


37° 


38° 


39° 


/ 




1 

2 
3 
4 

5 

6 

7 
8 
9 

10 

11 

12 
13 
14 

15 

16 
17 
18 
19 

20 

21 
22 
23 
24 

25 

26 
27 
28 
29 

30 

31 
32 
33 
34 

35 

36 
37 
38 
39 

40 

41 
42 
43 
44 

45 

.46 
47 
48 
49 

50 

51 
52 
53 

54 

55 

56 
57 
58 
59 

60 


tan 


cot 


tan 


cot 


tan 


cot 


tan 


cot 


tan 


cot 


60 

59 
58 
57 
56 

55 

54 
53 
52 
51 

50 

49 
48 
47 
46 
45 
44 
43 
42 
41 

40 

39 
38 
37 
36 

35 

34 
33 
32 
31 
30 

29 
28 
27 
26 

25 

24 
23 
22 
21 

20 

19 
18 
17 
16 
15 
14 
13 
12 
11 

10 

9 

8 
7 
6 

5 

4 
3 
2 
1 




7001! 


1.4281 


7265 


1.3764 


7536 


1.3270 


7813 


1.2799 


8098 


1.2349 


7006 
7011 
7015 
7019- 


1.4273 
1.4264 
1.4255 
1.4246 


7270 
7274 
7279 
7283 


1.3755 
1.3747 
1.3739 
1.3730 


7540 
7545 
7549 
7554 


1.3262 
1.3254 
1.3246 
1.3238 


7818 
7822 
7827 
7832 


1.2792 
1.2784 
1.2776 
1.2769 


8103 
8107 
8112 
8117 


1.2342 
1.2334 
1.2327 
1.2320 


7024 


1.4237 


7288 


1.3722 


7558 


1.3230 


7836 


1.2761 


8122 


1.2312 


7028 
7032 
7037 
7041 


1.4229 
1.4220 
1.4211 
1.4202 


7292 
7297 
7301 
7306 


1.3713 
1.3705 
1.3697 
1.3688 


7563 
7568 
7572 

7577 


1.3222 
1.3214 
1.3206 
1.3198 


7841 
7846 
7850 
7855 


1.2753 
1.2746 
1.2738 
1.2731 


8127 
8132 
8136 
8141 


1.2305 
1.2298 
1.2290 
1.2283 


7046 


1.4193 


7310 


1.3680 


7581 


1.3190 


7860 


1.2723 


8146 


1.2276 


7050 
7054 
7059 
7063 


1.4185 
1.4176 
1.4167 
1.4158 


7314 
7319 
7323 
7328 


1.3672 
1.3663 
1.3655 
1.3647 


7586 
7590 
7595 
7600 


1.3182 
1.3175 
1.3167 
1.3159 


7865 
7869 

7874 
7879 


1.2715 
1.2708 
1.2700 
1.2693 


8151 
8156 
8161 
8165 


1.2268 
1.2261 
1.2254 
1.2247 


7067 


1.4150 


7332 


1.3638 


7604 


1.3151 


7883 


1.2685 


8170 


1.2239 


7072 
7076 
7080 
7085 


1.4141 
1.4132 
1.4124 
1.4115 


7337 
7341 
7346 
7350 


1.3630 
1.3622 
1.3613 
1.3605 


7609 
7613 
7618 
7623 


1.3143 
1.3135 
1.3127 
1.3119 


7888 
7893 
7898 
7902 


1.2677 
1.2670 
1.2662 
1.2655 


8175 
8180 
8185 
8180 


1.2232 
1,2225 
1.2218 
1.2210 


7089 


1.4106 


7355 


1.3597 


7627 


1.3111 


7907 


1.2647 


8195 


1.2203 


7094 
7098 
7102 
7107 


1.4097 
1.4089 
1.4080 
1.4071 


7359 
7364 
7368 
7373 


1.3588 
1.3580 
1.3572 
1.3564 


7632 
7636 
7641 
7646 


1.3103 
1.3095 
1.3087 
1.3079 


7912 
7916 
7921 
7926 


1.2640 
1.2632 
1.2624 
1.2617 


8199 
8204 
8209 
8214 


1.2196 
1.2189 
1.2181 
1.2174 


7111 


1.4063 


7377 


1.3555 


7650 


1.3072 


7931 


1.2609 


8219 


1.2167 


7115 
7120 
7124 
7129 


1.4054 
1.4045 
1.4037 
1.4028 


7382 
7386 
7391 
7395 


1.3547 
1.3539 
1.3531 
1.3522 


7655 
7659 
7664 
7669 


1.3064 
1.3056 
1.3048 
1.3040 


7935 
7940 
7945 
7950 


1.2602 
1.2594 
1.2587 
1.2579 


8224 
8229 
8234 
8238 


1.2160 
1.2153 
1.2145 
1.2138 


7133 


1.4019 


7400 


1.3514 


7673 


1.3032 


7954 


1.2572 


8243 


1.2131 


7137 
7142 
7146 
7151 


1.4011 
1.4002 
1.3994 
1.3985 


7404 
7409 
7413 
7418 


1.3506 
1.3498 
1.3490 
1.3481 


7678 
7683 
7687 
7692 


1.3024 
1.3017 
1.3009 
1.3001 


7959 
7964 
7969 
7973 


1.2564 
1.2557 
1.2549 
1.2542 


8248 
8253 
8258 
8263 


1.2124 
1.2117 
1.2109 
1.2102 


7155 


1.3976 


7422 


1.3473 


7696 


1.2993 


7978 


1.2534 


8268 


1.2095 


7159 
7164 
7168 
7173 


1.3968 
1.3959 
1.3951 
1.3942 


7427 
7431 
7436 
7440 


1.3465 
1.3457 
1.3449 
1.3440 


7701 
7706 
7710 
7715 


1.2985 
1.2977 
1.2970 
1.2962 


7983 
7988 
7992 
7997 


1.2527 
1.2519 
1.2512 
1.2504 


8273 
8278 
8283 
8287 


1.2088 
1.2081 
1.2074 
1.2066 


7177 


1.3934 


7445 


1.3432 


7720 


1.2954 


8002 


1 . 2497 


8292 


1.2059 


7181 
7186 
7190 
7195 


1.3925 
1.3916 
1.3908 
1.3899 


7449 
7454 
7458 
7463 


1.3424 
1.3416 

1.3408 
1.3400 


7724 
7729 
7734 
7738 


1.2946 
1.2938 
1.2931 
1.2923 


8007 
8012 
8016 
8021 


1.2489 
1.2482 
1.2475 
1.2467 


8297 
8302 
8307 
8312 


1.2052 
1.2045 
1.2038 
1.2031 


7199 


1.3891 


7467 


1.3392 


7743 


1.2915 


8026 


1.2460 


8317 


1.2024 


7203 
7208 
7212 
7217 


1.3882 
1.3874 
1.3865 
1.3857 


7472 
7476 
7481 
7485 


1.3384 
1.3375 
1.3367 
1.3359 


7747 
7752 
7757 
7761 


1.2907 
1.2900 
1.2892 
1.2884 


8031 
8035 
8040 
8045 


1.2452 
1.2445 
1.2437 
1.2430 


8322 
8327 
8332 
8337 


1.2017 
1.2009 
1.2002 
1.1995 


7221 


1.3848 


7490 


1.3351 


7766 


1.2876 


8050 


1.2423 


8342 


1.1988 


7226 
7230 
7234 
7239 


1.3840 
1.3831 
1.3823 
1.3814 


7495 
7499 
7504 
7508 


1.3343 
1.3335 
1.3327 
1.3319 


7771 

7775 
7780 
7785 


1.2869 
1.2861 
1.2853 
1.2846 


8055 
8059 
8064 
8069 


1.2415 
1.2408 
1.2401 
1.2393 


8346 
8351 
8356 
8361 


1.1981 
1 . 1974 
1 . 1967 
1 . 1960 


7243 


1.3806 


7513 


1.3311 


7789 


1.2838 


8074 


1.2386 


8366 


1 . 1953 


7248 
7252 
7257 
7261 


1.3798 
1.3789 
1.3781 
1.3772 


7517 
7522 
7526 
7531 


1.3303 
1.3295 
1.3287 
1.3278 


7794 
7799 
7803 
7808 


1.2830 
1.2822 
1.2815 
1.2807 


8079 
8083 
8088 
8093 


1.2378 
1.2371 
1.2364 
1.2356 


8371 
8376 
8381 
8386 


1 . 1946 
1 . 1939 
1 . 1932 
1 . 1925 


7265 


1.3764 


7536 


1.3270 


7813 


1.2799 


8098 


1.2349 


8391 


1.1918 


cot 


tan 


cot 


tan 


cot 


tan 


cot 


tan 


cot 


tan 


/ 


54° 


53° 


52° 


51° 


50° 


/ 



9° 



NATURAL TANGENTS AND COTANGENTS 



! 


40° 


41° 


43° 


43° 


44° 


/ 




1 

2 
3 
4 

5 

6 
7 

8 
9 

10 

11 
12 
13 

14 

15 

16 
17 
18 
19 

30 

21 
22 
23 
24 

25 

26 
27 
28 
29 

30 

31 
32 
33 
34 

35 

36 
i 37 
I 38 

39 

40 

41 
42 
43 

44 

45 

I 46 

1 47 

48 

49 

50 

51 
52 
53 

54 

55 

, 56 

57 

: 58 

59 

60 


tan cot 


tan cot 


tan 


cot 


tan 


cot 


tan 


cot 


60 

59 ; 

58 
57 
56 

55 

54 
53 
52 
51 

50 

49 
48 
47 
46 
45 
44 
43 
42 
41 

40 

39 
38 
37 
36 

35 

34 
33 
32 
31 

30 

29 
28 
27 
26 

25 

24 
23 

22 
21 

20 

19 
18 
17 
16 
15 
14 
13 
12 
11 

10 

9- 
8 
7 
6 

5 

4 
3 
2 
1 




8391 1.1918 


8693 1 . 1504 

8698 1 . 1497 
8703 1 . 1490 
8708 1 . 1483 
8713 1.1477 


9004 


1.1106 


9325 


1.0724 


9657 


1.0355 


8396 1.1910 
8401 1.1903 
8406 1 . 1896 
8411 1.1889 


9009 
9015 
9020 
9025 


1.1100 
1 . 1093 
1 . 1087 
1 . 1080 


9331 
9336 
9341 
9347 


1.0717 
1.0711 
1.0705 
1.0699 


9663 
9668 
9674 
9679 


1.0349 
1.0343 
1.0337 
1.0331 


8416 1 . 1882 


8718 1 . 1470 


9030 


1.1074 


9352 


1.0692 


9685 


1.0325 


8421 1 . 1875 
8426 1 . 1868 
8431 1.1861 
8436 1 . 1854 


8724 1.1463 
8729 1 . 1456 
8734 1 . 1450 
8739 1.1443 


9036 
9041 
9046 
9052 


1 . 1067 
1.1061 
1 . 1054 
1 . 1048 


9358 
9363 
9369 
9374 


1.0686 
1.0680 
1.0674 
1.0668 


9691 
9696 
9702 
9708 


1.0319 
1.0313 
1.0307 
1.0301 


8441 1 . 1847 


8744 1 . 1436 
8749 1 . 1430 
8754 1 . 1423 
8759 1 . 1416 
8765 1.1410 


9057 


1.1041 


9380 


1.0661 


9713 


1.0295 


8446 1 . 1840 
8451 1.1833 
8456 1 . 1826 
8461 1 . 1819 


9062 
9067 
9073 
9078 


1.1035 
1 . 1028 
1 . 1022 
1.1016 


9385 
9391 
9396 
9402 


1.0655 
1.0649 
1.0643 
1.0637 


9719 
9725 
9730 
9736 


1.0289 
1.0283 
1.0277 
1.0271 


8466 1 . 1812 


8770 1 . 1403 


9083 


1 . 1009 


9407 


1.0630 


9742 


1.0265 


8471 1 . 1806 
8476 1 . 1799 
8481 1.1792 
8486 1 . 1785 


8775 1 . 1396 
8780 1.1389 
8785 1 . 1383 
8790 1 . 1376 


9089 
9094 
9099 
9105 


1 . 1003 
1.0996 
1.0990 
1.0983 


9413 
9418 
9424 
9429 


1.0624 
1.0618 
1.0612 
1.0606 


9747 
9753 
9759 
9764 


1.0259 
1.0253 
1.0247 
1.0241 


8491 1.1778 


8796 1.1369 


9110 


1.0977 


9435 


1 .0599 


9770 


1.0235 


8496 1 . 1771 
8501 1 . 1764 
8506 1.1757 
8511 1.1750 


8801 1 . 1363 
8806 1 . 1356 
8811 1.1349 
8816 1.1343 


9115 
9121 
9126 
9131 


1.0971 
1.0964 
1.0958 
1.0951 


9440 
9446 
9451 
9457 


1.0593 
1.0587 
1.0581 
1.0575 


9776 
9781 
9787 
9793 


1.0230 
1.0224 
1.0218 
1.0212 


8516 1 . 1743 


8821 1 . 1336 


9137 


1.0945 


9462 


1.0569 


9798 


1.0206 


8521 1.1736 
8526 1 . 1729 
8531 1.1722 
8536 1.1715 


8827 1.1329 
8832 1.1323 
8837 1.1316 
8842 1 . 1310 


9142 
9147 
9153 
9158 


1.0939 
1.0932 
1.0926 
1.0919 


9468 
9473 
9479 
9484 


1.0562 
1.0556 
1.0550 
1.0544 


9804 
9810 
9816 
9821 


1.0200 
1.0194 
1.0188 
1.0182 


8541 1 . 1708 


8847 1 . 1303 


9163 


1.0913 


9490 


1.0538 


9827 


1.0176 


8546 1 . 1702 
8551 1.1695 
8556 1 . 1683 
8561 1 . 1681 


8852 1 . 1296 
8858 1 . 1290 
8863 1 . 1283 
8868 1 . 1276 


9169 
9174 
9179 
9185 


1.0907 
1.0900 
1.0894 
1.0888 


9495 
9501 
9506 
9512 


1.0532 
1.0526 
1.0519 
1.0513 


9833 
9838 
9844 
9850 


1.0170 
1.0164 
1.0158 
1.0152 


8566 1 . 1674 


8873 1 . 1270 


9190 


1.0881 


9517 


1.0507 


9856 


1.0147 


8571 1.1667 
8576 1 . 1660 
8581 1 . 1653 
8586 1 . 1647 


8878 1 . 1263 
8884 1 . 1257 
8889 1.1250 
8894 1 . 1243 


9195 
9201 
9206 
9212 


1.0875 
1.0869 
1.0862 
1.0856 


9523 
9528 
9534 

9540 


1.0501 
1.0495 
1.0489 
1.0483 


9861 
9867 
9873 
9879 


1.0141 
1.0135 
1.0129 
1.0123 


8591 1.1640 


8899 1 . 1237 


9217 


1.0850 


9545 


1.0477 


9884 


1.0117 


8596 1.1633 
8601 1 . 1626 
8606 1 . 1619 
8611 1.1612 


8904 1 . 1230 
8910 1.1224 
8915 1.1217 
8920 1.1211 


9222 
9228 
9233 
9239 


1.0843 
1.0837 
1.0831 
1.0824 


9551 
9556 
9562 
9567 


1.0470 
1.0464 
1.0458 
1.0452 


9890 
9896 
9902 
9907 


1.0111 
1.0105 
1.0099 
1.0094 


8617 1 . 1606 


8925 1 . 1204 


9244 


1.0818 


9573 


1.0446 


9913 


1.0088 


8622 1 . 1599 
8627 1 . 1592 
8632 1 . 1585 
8637 1 . 1578 


8931 1.1197 
8936 1.1191 
8941 1.1184 
8946 1.1178 


9249 
9255 
9260 
9266 


1.0812 
1.0805 
1.0799 
1.0793 


9578 
9584 
9590 
9595 


1.0440 
1.0434 
1.0428 
1.0422 


9919 
9925 
9930 
9936 


1.0082 
1.0076 
1.0070 
1.0064 


8642 1 . 1571 


8952 1.1171 


9271 


1.0786 


9601 


1.0416 


9942 

9948 
9954 
9959 
9965 


1.0058 
1.0052 
1.0047 
1.0041 
1.0035 


8647 1.1565 
8652 1 . 1558 
8657 1 . 1551 
8662 1 . 1544 


8957 1.1165 
8962 1.1158 
8967 1.1152 
8972 1.1145 


9276 
9282 
9287 
9293 


1.0780 
1.0774 
1.0768 
1.0761 


9606 
9612 
9618 
9623 


1.0410 
1.0404 
1.0398 
1.0392 


8667 1 . 1538 


8978 1.1139 


9298 


1.0755 


9629 


1.0385 


9971 


1.0029 


8672 1 . 1531 
8678 1.1524 
8683 1.1517 
8688 1.1510 


8983 1.1132 
8988 1.1126 
8994 1.1119 
8999 1.1113 


9303 
9309 
9314 
9320 

9325 


1.0749 
1.0742 
1.0736 
1.0730 

1.0724 


9634 
9640 
9646 
9651 


1.0379 
1.0373 
1.0367 
1.0361 


9977 
9983 
9988 
9994 

1.000 


1.0023 
1.0017 
1.0012 
1.0006 
1.0000 


8693 1 . 1504 


9004 1.1106 


9657 


1.0355 




cot tan 


cot tan 


cot 


tan 


cot 


tan 


cot 


tan 




1 49° 


48° 


47° 


46° 


45° 


/ 



CONSTANTS 



9 1 



TABLE V.— CONSTANTS. 



Radians in one degree 
Radians in one minute 
Radians in one second 
sin i" 
tan i" 

Degrees in one radian 
Minutes in one radian 
Seconds in one radian 
ir (Ratio of circum. to diam.) 
i 
x 

7T 2 



7T Z _ 

vV 



Vx 

M (Modulus of common logarithms) 

e (Napierian base) 

i 
e 



V2 

•Si 

1 meter 

1 kilometer 

1 foot 

1 mil* 

1 mile 

1 nautical mile 

Feet per sec. in 1 mile per hour 

Miles per hour in 1 foot per second 

360 



Number. 
0-017 453 3 
O.000 290 9 
0.000 004 8 
0.000 004 8 
0.000 004 8 

57-295 779 5 
3 437-740 77 
206 264.806 

3. 141 592 7 

0.318 309 9 
9.869 604 4 



Logarithm. 
8.241 877 4 
6.463 726 I 
4.685 574 9 
4.685 574 9 
4.685 574 9 
1.758 122 6 
3-536 273 9 
5.314425 1 
0.497 149 9 

9.502 850 1 

0.994 299 7 



0.101 321 2 9.005 700 3 
1-772 453 9 0.248 574 9 



0.564 189 6 

0-434 294 5 
2.718 281 8 

0.367 879 4 

7.389 005 6 

o.i35 335 3 

2 

1.414 213 6 

3 

1.732 050 8 

5 

2.236 068 o 
■■ 39.370 000 inches 
: 3.280 833 feet 
0.621 370 mile 
0.304 801 meter 
1.609 347 kilom. 

5280 feet 

6080.290 feet 

1.466 667 

0.681 818 

: 21 600' 

I 296 OOO" 



9.751 425 I 

9.637 7843 
O.434 294 5 

9-565 705 5 

0.868 589 o 

9. 131 411 o 

0.301 030 o 
0.150 515 o 

O.477 121 3 

0.238 560 6 
0.698 970 o 
0.369 485 o 
1.595 165 4 
0.515 984 2 

9-793 3503 
9.484 015 8 
0.206 649 7 
3.722 633 9 
3.783 9244 
0.166 331 5 
9.833 368 6 

4-334 453 8 
6. 112 605 o 



INDEX 

(References refer to the pages) 



Abscissa, I, 117 

Absolute value, I, 281 

Absolutely convergent series, I, 308 

Accuracy of results, I, 43 

Acute spherical triangles, II, 3 

Addition theorems, 

cosine, I, 210, 211, 215 

sine, I, 210, 211 

tangent, I, 215 
Adjacent parts, II, 17 
Ambiguous case, 

of oblique spherical triangles, II, 10, 
n, 58, 60 

of plane triangles, I, 141 

of right spherical triangles, II, 21 
Amplitude, I, 257, 281 
Angle, definition of, I, 177 

measure of, I, 179, 181 

of depression, I, 5 

of elevation, I, 5 
Antilogarithms, I, 75 
Archimedes, I, 182 
Arc sine, I, 18 
Arc tangent, I, 18 
Argand, J. R., I, 285 . 
Argument, I, 281 
Arithmetic complement, I, 56 
Asymptote, I, 258 
Auxiliary angle, I, 240 

Binomial series, I, 314 

Biquadrantal spherical triangles, II, 3 

Birectangular spherical triangles, II, 4 

Bowditch's rules, II, 31 

Briggsian logarithms, I, 57 

Buergi, Jost, I, 87 

Cagnoli's formula, II, 46 
Catenary, I, 274 



Cauchy, I, 285 
Centesimal measure, I, 180 
Characteristic of logarithms, I, 58 

rules for, I, 59 
Circular functions, I, 10 

measure, I, 181 

parts, II, 17 
Cof unctions, defined, I, 19 
Cologarithms, I, 56 
Co-lunar triangles, defined, II, 4 

use of, II, 4 
Common logarithms defined, I, 57 

tables of, I, 61 
Complement arithmetic, I, 56 

of an angle, I, 178 
Complex numbers, defined, I, 280 

trigonometric form of, I, 281 
Compound interest law, I, 272 
Conditionally convergent series, I, 308 
Conjugate complex numbers, I, 288 
Convergency test, I, 311 
Convergent series, I, 307 
Coordinates, I, 117 
Cosecant, I, 10, 118, 191, 338 

hyperbolic, I, 342 
Cosine, I, 10, 118, 192, 338 

curve, I, 260 

hyperbolic, I, 342 

law of, I, 128 

law of spherical, II, 34 

of difference, I, 210 

of double angle, I, 216 

of half angle, I, 216 

of sum, I, 210, 211 

of 18 , I, 24, 218 

series, I, 313, 329 
Cotangent, I, 10, 118, 192, 338 

hyperbolic, I, 342 
Coterminal angles, I, 178 



93 



94 



INDEX 



Co versed sine, I, 10 
Cyclic substitution, I, 127 

Damped vibrations, curve of, I, 275 
Decimal measure of angles, I, 180 
Delambre's proportions, II, 43 
De Moivre, I, 289. 
De Moivre's theorem, I, 289 
Departure, denned, I, 121, 270 
Depression, angle of, I, 5 
Descartes, I, 285 
Divergent series, I, 307 
Double formula, I, 131, 223 

Elevation, angle of, I, 5 
Equilateral spherical triangle, II, 3 
Euler's theorem, I, 338 
Exponential curves, I, 270 

equations, I, 71 

series, I, 313, 318 

Fourier's theorem, I, 268 
Frequency of oscillation, I, 263 
Function, definition of, I, 9 

of an acute angle, I, 9 

of an obtuse angle, I, 118 

of any angle, I, 191 

of imaginary angles, I, 337 
Fundamental laws of logarithms, I, 54 

relations, I, 24, 119 

Gauss, I, 285 

Gauss's proportions, II, 43 
General spherical triangles, II, 11 
Geometric series, I, 310 
Goniometric functions, I, 10 
Graphic solution, 

of plane triangles, I, 3 

of spherical triangles, II, 7 

Half -angle formulas, 

for plane triangles, I, 134 
for spherical triangles, II, 40 

Half-side formulas, II, 41 

Hansen's problem, I, 150 

Harmonic curves, I, 260 



Hero of Alexandria, I, 133 

Hipparchus, I, 7, 50 

Hyperbola, I, 255 

Hyperbolic functions, defined, I, 342 

formulas, I, 345 

functions, curves of, I, 277 

logarithms, I, 76 

sector, area of, I, 350 

Identities, I, 32 
Imaginary numbers, I, 278 

unit, I, 278 
Infinite series, definition of, I, 306 
Interpolation, I, 37 
Inverse functions, I, 245 

hyperbolic functions, I, 348 

sine, I, 17 

tangent, I, 17 
Isosceles spherical triangles, II, 3 

Laplace, I, 53 

Latitude, defined, I, 101, 170 
Lhuilier's formula, II, 46 
Logarithm, defined, I, 53 
Logarithmic curves, I, 269 

decrement, I, 276 

series, I, 314, 320 

trigonometric functions, I, 78 
Logarithms, applications of, I, 72 

characteristic of, I, 58 

computation of, I, 73, 321. 

directions for use of, I, 67 

mantissa of, I, 58 
Ludolph van Ceulen, I, 182 

Mantissa, I, 58 
Mariner's compass, I, 101 
Middle parts, II, 17 
Modo-cyclic functions, I, 343 
Modulus of logarithms, I, 76, 323 

of complex numbers, I, 281 
Mollweide, I, 131 

Napier, John, I, 54, 76, 87 
Napierian logarithms, I, 76 
Napier's proportions, II, 44 
circular parts, II, 17, 18 



INDEX 



95 



Natural functions, 

computation of, I, 330 

tables of, I, 14, 35 
Natural logarithms, I, 76 

system of angular measure, I, 181 
Newton, I, 131 
Non-convergent series, I, 307 

Oblique spherical triangles, II, 3 
Obtuse spherical triangles, II, 3 
Opposite parts, II, 18 
Ordinate, I, 117 
Origin, I, 117 
Oscillating series, I, 307 

Period, I, 193, 257 

of oscillation, I, 263 
Periodic curves, I, 257 

functions, I, 193, 257 

time, I, 263 
Periodicity of trigonometric functions, 

I, 192 
Pitiscus, I, 51 
Polar triangle, definition of, II, 6 

use of, II, 8 
Pothenot's problem, I, 153 
Principle value of angle, I, 178, 225 
Projection, defined, I, 127 

theorem, I, 126 
Proportional parts, principle of, I, 38 
Ptolemaus, I, 50 

Quadrantal spherical triangles, II, 3 

Radian, I, 181 

measure, I, 181 
Ratio of convergence, I, 312 
Reciprocal, denned, I, 9 

relations, I, 24 
Rectangular coordinates, I, 117 
Reflection of a curve, I, 272 
Rheticus, I, 50 
Right spherical triangles, II, 3 

ambiguous case of, II, 21 

Scalene spherical triangles, II, 3 
Schultze, J. H., I, 180 



Secant, defined, I, 10, 118, 192, 338 

hyperbolic, I, 342 
Semi-convergent series, I, 308 
Series, defined, I, 306 

absolutely convergent, I, 308 

binomial, I, 314 

conditionally convergent, I, 308 

convergent, I, 307 

cosine, I, 313, 329 

divergent, I, 307 

exponential, I, 313, 318 

geometric, I, 310 

logarithmic, I, 314, 320 

non-convergent, I, 307 

oscillating, I, 307 

sine, I, 313, 329 

tangent, I, 329 

with complex terms, I, 336 
Significant figures, I, 93 
Simple harmonic curves, I, 260 

motion, I, 263 
Sine, defined, I, 10, 118, 191, 338 

curve, I, 255 

hyperbolic, I, 342 

of difference, I, 210 

of double angle, I, 216 

of half angle, I, 216 

of 18°, I, 24, 218 

of sum, I, 210, 211 

series, I, 313, 329 
Sines, law of, 

for plane triangles, I, 126 

for spherical triangles, II, 33 
Sinusoidal curves, I, 260 
Small angles, 

functions of, I, 44, 83 

in plane triangles, I, 49 

in spherical triangles, II, 26 
Snellius, I, 153 
Spherical triangles, area of, II, 45 

classification of, II, 3 

co-lunar, II, 4 

general, II, n 

polar, II, 6 
Spherical trigonometry, II, 1 
Spheroidal trigonometry, II, 2 
Square relations, I, 24 



DEC 19 1913 



96 

Subtraction theorems, I, 210, 216 
Supplement of an angle, I, 178 
S and T tables, I, 84 

Tables, of common logarithms, I, 61 

of log. trig, functions, I, 78 

of natural functions, I, 35 

S and T, I, 84 
Tabular logarithmic sine, I, 79 
Tangent, defined, I, 10, 118, 192, 338 

curve, I, 257 

hyperbolic, I, 342 

law of tangents, I, 131, 223 

of difference, I, 216 

of double angle, I, 216 

of half angle, I, 217 

of 18 , 1, 24 



INDEX 



Tangent, of sum, I, 215 

series, I, 329 
Test ratio, I, 312 
Three-point problem, I, 150, 223 
Trigonometric functions, I, 10, 118, 337 

formulas, I, 345 
Trigonometry, defined, 

plane, I, 8 

spherical, II, 1 
Triquadrantal spherical triangles, II, 3 
Trirectangular spherical triangles, II, 4 

Versed sine, I, 10 
Vlacq, Adrian, I, 87 

Wave length, I, 257 
Wessel, Caspar, I, 285 



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